Polynomials

Definition of a Polynomial

A polynomial is an expression that can be written as a term or a sum of more than one term of the form $ ax_{1}{n_{1}}x_{2}{n_{2}}x_{m}^{n_{m}} $ , where the a is a constant and the $ x_{1},,x_{m} $ are variables. A polynomial of one term is called a monomial. A polynomial of two terms is called a binomial. A polynomial of three terms is called a trinomial.

EXAMPLE 2.1 5, -20, $ $ , t, $ 3x^{2} $ , $ -15x{3}y{2} $ , $ xy^{4}zw $ are monomials.

EXAMPLE 2.2 $ x + 5, x^{2} - y^{2}, 3x{5}y{7} - x^{3}z $ are binomials.

EXAMPLE 2.3 $ x + y + 4z $ , $ 5x^{2} - 3x + 1 $ , $ x^{3} - y^{3} + t^{3} $ , $ 8xyz - 5x^{2}y + 20t^{3}u $ are trinomials.

The Degree of a Term

The degree of a term in a polynomial is the exponent of the variable, or, if more than one variable is present, the sum of the exponents of the variables. If no variables occur in a term, it is called a constant term. The degree of a constant term is 0.

EXAMPLE 2.4 (a) $ 3x^{8} $ has degree 8; (b) $ 12xy{2}z{2} $ has degree 5; (c) $ $ has degree 0.

The Degree of a Polynomial

The degree of a polynomial with more than one term is the largest of the degrees of the individual terms.

EXAMPLE 2.5 (a) $ x^{4} + 3x^{2} - 250 $ has degree 4; (b) $ x{3}y{2} - 30x^{4} $ has degree 5; (c) $ 16 - x - x^{10} $ has degree 10; (d) $ x^{3} + 3x^{2}h + 3xh^{2} + h^{3} $ has degree 3.

Like and Unlike Terms

Two or more terms are called like terms if they are both constants, or if they contain the same variables raised to the same exponents, and differ only, if at all, in their constant coefficients. Terms that are not like terms are called unlike terms.

EXAMPLE 2.6 3x and 5x, $ -16x^{2}y $ and $ 2x^{2}y $ , $ tu^{5} $ and $ 6tu^{5} $ are examples of like terms. 3 and 3x, $ x^{2} $ and $ y^{2} $ , $ a{3}b{2} $ and $ a{2}b{3} $ are examples of unlike terms.

Addition

The sum of two or more polynomials is found by combining like terms. Order is unimportant, but polynomials in one variable are generally written in order of descending degree in their terms. A polynomial in one variable, x, can always be written in the form:

\[ a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0} \]

This form is generally referred to as standard form. The degree of a polynomial written in standard form is immediately seen to be n.

EXAMPLE 2.7

\[ 5x^{3}+6x^{4}-8x+2x^{2}=6x^{4}+5x^{3}+2x^{2}-8x\ (degree4) \]

EXAMPLE 2.8

\[ \begin{aligned}(x^{3}-3x^{2}+8x+7)+(-5x^{3}-12x+3)&=x^{3}-3x^{2}+8x+7-5x^{3}-12x+3\\&=-4x^{3}-3x^{2}-4x+10\end{aligned} \]

Subtraction

The difference of two polynomials is found using the definition of subtraction: $ A - B = A + (-B) $ . Note that to subtract B from A, write A - B.

EXAMPLE 2.9

\[ \begin{aligned}(y^{2}-5y+7)-(3y^{2}-5y+12)&=(y^{2}-5y+7)+(-3y^{2}+5y-12)\\&=y^{2}-5y+7-3y^{2}+5y-12\\&=-2y^{2}-5\end{aligned} \]

Multiplication

The product of two polynomials is found using various forms of the distributive property as well as the first law of exponents: $ x{a}x{b}=x^{a+b} $

EXAMPLE 2.10

\[ \begin{aligned}x^{3}(3x^{4}-5x^{2}+7x+2)&=x^{3}\cdot3x^{4}-x^{3}\cdot5x^{2}+x^{3}\cdot7x+x^{3}\cdot2\\&=3x^{7}-5x^{5}+7x^{4}+2x^{3}\end{aligned} \]

EXAMPLE 2.11 Multiply: $ (x + 2y)(x^{3} - 3x^{2}y + xy^{2}) $

\[ \begin{aligned}(x+2y)(x^{3}-3x^{2}y+xy^{2})&=(x+2y)x^{3}-(x+2y)3x^{2}y+(x+2y)xy^{2}\\&=x^{4}+2x^{3}y-3x^{3}y-6x^{2}y^{2}+x^{2}y^{2}+2xy^{3}\\&=x^{4}-x^{3}y-5x^{2}y^{2}+2xy^{3}\end{aligned} \]

Often a vertical format is used for this situation:

\[ \begin{aligned}&x^{3}-3x^{2}y+xy^{2}\\&x+2y\\&\overline{x^{4}-3x^{3}y+x^{2}y^{2}}\\&\frac{2x^{3}y-6x^{2}y^{2}+2xy^{3}}{x^{4}-x^{3}y-5x^{2}y^{2}+2xy^{3}}\\ \end{aligned} \]

The FOIL (First Outer Inner Last) Method

The FOIL (First Outer Inner Last) method for multiplying two binomials:

\[ (a+b)(c+d)=ac+ad+bc+bd \]

First Outer Inner Last

EXAMPLE 2.12

\[ (2x+3)(4x+5)=8x^{2}+10x+12x+15=8x^{2}+22x+15 \]

Special Product Forms

\[ (a+b)(a-b)=a^{2}-b^{2} \]

\[ (a+b)^{2}=(a+b)(a+b)=a^{2}+2ab+b^{2} \]

\[ (a-b)^{2}=(a-b)(a-b)=a^{2}-2ab+b^{2} \]

Difference of two squares

Square of a sum

Square of a difference

\[ \begin{aligned}&(a-b)(a^{2}+ab+b^{2})=a^{3}-b^{3}\quad&Difference of two cubes\\&(a+b)(a^{2}-ab+b^{2})=a^{3}+b^{3}\quad&Sum of two cubes\ $ a+b)^{3}=(a+b)(a+b)^{2}\quad&Cube of a sum\\&=(a+b)(a^{2}+2ab+b^{2})=a^{3}+3a^{2}b+3ab^{2}+b^{3}\ $ a-b)^{3}&=(a-b)(a-b)^{2}\quad&Cube of a difference\\&=(a-b)(a^{2}-2ab+b^{2})=a^{3}-3a^{2}b+3ab^{2}-b^{3}\end{aligned} \]

Factoring

Factoring polynomials reverses the distributive operations of multiplication. A polynomial that cannot be factored is called prime. Common factoring techniques include: removing a common factor, factoring by grouping, reverse FOIL factoring, and special factoring forms.

EXAMPLE 2.13 Removing a monomial common factor: $ 3x{5}-24x{4}+12x{3}=3x{3}(x^{2}-8x+4) $

EXAMPLE 2.14 Removing a nonmonomial common factor:

\[ \begin{aligned}12(x^{2}-1)^{4}(3x+1)^{3}+8x(x^{2}-1)^{3}(3x+1)^{4}&=4(x^{2}-1)^{3}(3x+1)^{3}[3(x^{2}-1)+2x(3x+1)]\\&=4(x^{2}-1)^{3}(3x+1)^{3}(9x^{2}+2x-3)\end{aligned} \]

It is important to note that the common factor in such problems consists of each base to the lowest exponent present in each term.

EXAMPLE 2.15 Factoring by grouping:

\[ 3x^{2}+4xy-3xt-4ty=(3x^{2}+4xy)-(3xt+4ty)=x(3x+4y)-t(3x+4y)=(3x+4y)(x-t) \]

Reverse FOIL factoring follows the patterns:

\[ \begin{aligned}x^{2}+(a+b)x+ab&=(x+a)(x+b)\\acx^{2}+(bc+ad)xy+bdy^{2}&=(ax+by)(cx+dy)\end{aligned} \]

EXAMPLE 2.16 Reverse FOIL factoring:

  1. To factor $ x^{2}-15x+50 $ , find two factors of 50 that add to -15: -5 and -10.

\[ x^{2}-15x+50=(x-5)(x-10) \]

  1. To factor $ 4x^{2} + 11xy + 6y^{2} $ , find two factors of $ 4 = 24 $ that add to 11:8 and 3.

\[ 4x^{2}+11xy+6y^{2}=4x^{2}+8xy+3xy+6y^{2}=4x(x+2y)+3y(x+2y)=(x+2y)(4x+3y) \]

Special Factoring Forms

\[ a^{2}-b^{2}=(a+b)(a-b) \]

Difference of two squares

\[ a^{2}+b^{2}is prime. \]

Sum of two squares

\[ a^{2}+2ab+b^{2}=(a+b)^{2} \]

Square of a sum

\[ a^{2}-2ab+b^{2}=(a-b)^{2} \]

Square of a difference

\[ a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2}) \]

Sum of two cubes

\[ a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2}) \]

Difference of two cubes

General Factoring Strategy

Step 1: Remove all factors common to all terms.

Step 2: Note the number of terms.

If the polynomial remaining after step 1 has two terms, look for a difference of two squares, or a sum or difference of two cubes.

If the polynomial remaining after step 1 has three terms, look for a perfect square or try reverse FOIL factoring. If the polynomial remaining after step 1 has four or more terms, try factoring by grouping.

SOLVED PROBLEMS

2.1. Find the degree of: (a) 12; (b) $ 35x^{3} $ ; (c) $ 3x^{3} - 5x^{4} + 3x^{2} + 9 $ ; (d) $ x^{8} - 64 $

  1. This polynomial has one term and no variables. The degree is 0.

  2. This polynomial has one term. The exponent of the variable is 3. The degree is 3.

  3. This polynomial has four terms, of degrees 3,4,2,0, respectively. The largest of these is 4, hence the degree of the polynomial is 4.

  4. This polynomial has two terms, of degrees 8 and 0, respectively. The largest of these is 8, hence the degree of the polynomial is 8.

2.2. Find the degree of (a) $ x^{2}y $ (b) $ xy - y^{3} + 7 $ (c) $ x^{4} + 4x^{3}h + 6x{2}h{2} + 4xh^{3} + h^{4} $

  1. This polynomial has one term. The sum of the exponents of the variables is $ 2 + 1 = 3 $ , hence the degree of the polynomial is 3.

  2. This polynomial has three terms, of degrees 2,3,0, respectively. The largest of these is 3, hence the degree of the polynomial is 3.

  3. This polynomial has five terms, each of degree 4, hence the degree of the polynomial is 4.

2.3. If $ A = x^{2} - 6x + 10 $ and $ B = 3x^{3} - 7x^{2} + x + 1 $ , find (a) $ A + B $ (b) A - B.

\[ \begin{aligned}A+B&=(x^{2}-6x+10)+(3x^{3}-7x^{2}+x+1)\\&=x^{2}-6x+10+3x^{3}-7x^{2}+x+1\\&=3x^{3}-6x^{2}-5x+11\end{aligned} \]

\[ \begin{aligned}A-B&=(x^{2}-6x+10)-(3x^{3}-7x^{2}+x+1)\\&=x^{2}-6x+10-3x^{3}+7x^{2}-x-1\\&=-3x^{3}+8x^{2}-7x+9\end{aligned} \]

2.4. Add $ 8x{3}-y{3} $ and $ x{2}-5xy{2}+y^{3} $ .

\[ (8x^{3}-y^{3})+(x^{2}-5xy^{2}+y^{3})=8x^{3}-y^{3}+x^{2}-5xy^{2}+y^{3}=8x^{3}+x^{2}-5xy^{2} \]

2.5. Subtract $ 8x{3}-y{3} $ from $ x{2}-5xy{2}+y^{3} $

\[ (x^{2}-5xy^{2}+y^{3})-(8x^{3}-y^{3})=x^{2}-5xy^{2}+y^{3}-8x^{3}+y^{3}=-8x^{3}+x^{2}-5xy^{2}+2y^{3} \]

2.6. Simplify: $ 3x{2}-5x-(5x+8-(8-5x{2}+(3x^{2}-x+1))) $

\[ \begin{aligned}3x^{2}-5x-(5x+8-(8-5x^{2}+(3x^{2}-x+1)))&=3x^{2}-5x-(5x+8-(8-5x^{2}+3x^{2}-x+1))\\&=3x^{2}-5x-(5x+8-(-2x^{2}-x+9))\\&=3x^{2}-5x-(5x+8+2x^{2}+x-9)\\&=3x^{2}-5x-(2x^{2}+6x-1)\\&=x^{2}-11x+1\end{aligned} \]

2.7. Multiply: (a) $ 12x{2}(x{2}-xy+y^{2}) $ ; (b) $ (a+b)(2a-3) $ ; (c) $ (3x-1)(4x^{2}-8x+3) $

  1. $ 12x{2}(x{2}-xy+y{2})=12x{2}x{2}-12x{2}xy+12x^{2}y{2}=12x{4}-12x{3}y+12x{2}y^{2} $

  2. $ (a + b)(2a - 3) = a(2a - 3) + b(2a - 3) $

\[ = 2a^{2} - 3a + 2ab - 3b \]

\[ \begin{aligned}(c)\ (3x-1)(4x^{2}-8x+3)&=(3x-1)4x^{2}-(3x-1)8x+(3x-1)3\\&=12x^{3}-4x^{2}-24x^{2}+8x+9x-3\\&=12x^{3}-28x^{2}+17x-3\end{aligned} \]

2.8. Multiply, using the vertical scheme: \((4p - 3q)(2p^{3} - p^{2}q + pq^{2} - 2q^{3})\)

\[ 2p^{3}\;-\quad p^{2}q\;+\;p q^{2}\quad-2q^{3} \]

\[ \begin{array}{r} 4p-3q\end{array} \]

\[ 8p^{4}-4p^{3}q+4p^{2}q^{2}-8p q^{3} \]

\[ -6p^{3}q+3p^{2}q^{2}-3pq^{3}+6q^{4} \]

\[ 8p^{4}-10p^{3}q+7p^{2}q^{2}-11p q^{3}+6q^{4} \]

2.9. Multiply:

  1. $ (cx - d)(cx + d) $ ; (b) $ (3x - 5)^{2} $ ; (c) $ (2t - 5)(4t^{2} + 10t + 25) $ ;

  2. $ 4(-2x)(1 - x{2}){3} $ ; (e) $ [(r - s) + t][(r - s) - t] $

  3. $ (cx - d)(cx + d) = (cx)^{2} - d^{2} = c{2}x{2} - d^{2} $

  4. $ (3x - 5)^{2} = (3x)^{2} - 2(3x) + 5^{2} = 9x^{2} - 30x + 25 $

  5. \((2t-5)(4t^{2}+10t+25)=(2t)^{3}-5^{3}=8t^{3}-125\) using the difference of two cubes pattern.

\[ \begin{aligned}(d)\ 4(-2x)(1-x^{2})^{3}&=-8x(1-x^{2})^{3}\\&=-8x(1-3x^{2}+3x^{4}-x^{6})\\&=-8x+24x^{3}-24x^{5}+8x^{7}\end{aligned}\quad using the cube of a difference pattern. \]

  1. $ [(r-s)+t][(r-s)-t]=(r-s){2}-t{2}=r{2}-2rs+s{2}-t^{2} $ using the difference of two squares pattern, followed by the square of a difference pattern.

2.10. Perform indicated operations: (a) $ (x + h)^{3} - (x - h)^{3} $ ; (b) $ (1 + t)^{4} $ .

\[ \begin{aligned}(b)\ (1+t)^{4}&=((1+t)^{2})^{2}=(1+2t+t^{2})^{2}=(1+2t)^{2}+2(1+2t)t^{2}+t^{4}\\&=1+4t+4t^{2}+2t^{2}+4t^{3}+t^{4}=1+4t+6t^{2}+4t^{3}+t^{4}\end{aligned} \]

2.11. Factor: (a) $ 15x^{4} - 10x^{3} + 25x^{2} $ ; (b) $ x^{2} + 12x + 20 $ ; (c) $ 9x^{2} - 25y^{2} $ ;

  1. $ 6x{5}-48x{4}-54x^{3} $ ; (e) $ 5x{2}+13xy+6y{2} $ ; (f) $ P(1+r)+P(1+r)r $ ; (g) $ x^{3}-64 $ ;

  2. $ 3(x + 3)^{2}(x - 8)^{4} + 4(x + 3)^{3}(x - 8)^{3} $ ; (i) $ x^{4} - y^{4} + x^{3} - xy^{2} $ ; (j) $ x^{6} - 64y^{6} $

  3. $ 15x{4}-10x{3}+25x{2}=5x{2}(3x^{2}-2x+5) $ . After removing the common factor, the remaining polynomial is prime.

  4. $ x^{2} + 12x + 20 = (x + 10)(x + 2) $ using reverse FOIL factoring.

  5. $ 9x{2}-25y{2}=(3x){2}-(5y){2}=(3x-5y)(3x+5y) $ using the difference of two squares pattern.

  6. $ 6x{5}-48x{4}-54x{3}=6x{3}(x{2}-8x-9)=6x{3}(x-9)(x+1) $ removing the common factor, then using reverse FOIL factoring.

  7. $ 5x^{2} + 13xy + 6y^{2} = (5x + 3y)(x + 2y) $ using reverse FOIL factoring.

  8. $ P(1 + r) + P(1 + r)r = P(1 + r)(1 + r) = P(1 + r)^{2} $ . Here, the common factor $ P(1 + r) $ was removed from both terms.

  9. $ x^{3} - 64 = (x - 4)(x^{2} + 4x + 16) $ using the difference of two cubes pattern.

  10. Removing the common factor from both terms and combining terms in the remaining factor yields:

\[ \begin{aligned}3(x+3)^{2}(x-8)^{4}+4(x+3)^{3}(x-8)^{3}&=(x+3)^{2}(x-8)^{3}\left[3(x-8)+4(x+3)\right]\\&=(x+3)^{2}(x-8)^{3}\left(7x-12\right)\end{aligned} \]

\[ \begin{aligned}(i)\quad x^{4}-y^{4}+x^{3}-xy^{2}&=(x^{4}-y^{4})+(x^{3}-xy^{2})\\&=(x^{2}-y^{2})(x^{2}+y^{2})+x(x^{2}-y^{2})\\&=(x^{2}-y^{2})(x^{2}+y^{2}+x)\\&=(x-y)(x+y)(x^{2}+y^{2}+x)\end{aligned} \]

\[ \begin{aligned}(j)\quad&x^{6}-64y^{6}=(x^{3}-8y^{3})(x^{3}+8y^{3})=(x-2y)(x^{2}+2xy+4y^{2})(x+2y)(x^{2}-2xy+4y^{2})\end{aligned} \]

2.12. A special factoring technique that is occasionally of use involves adding a term to make a polynomial into a perfect square, then subtracting that term immediately. If the added term is itself a perfect square, then the original polynomial can be factored as the difference of two squares. Illustrate this technique for (a) $ x^{4} + 4y^{4} $ ; (b) $ x^{4} + 2x{2}y{2} + 9y^{4} $ .

  1. Since $ x^{4} + 4y^{4} = (x{2}){2} + (2y{2}){2} $ , adding $ 2x{2}(2y{2}) = 4x{2}y{2} $ makes the polynomial into a perfect square. Then subtracting this quantity yields a difference of two squares, which can be factored:

\[ \begin{aligned}x^{4}+4y^{4}&=x^{4}+4x^{2}y^{2}+4y^{4}-4x^{2}y^{2}\\&=(x^{2}+2y^{2})^{2}-(2xy)^{2}\\&=(x^{2}+2y^{2}-2xy)(x^{2}+2y^{2}+2xy)\\ \end{aligned} \]

  1. If the middle term of this polynomial were $ 6x{2}y{2} $ instead of $ 2x{2}y{2} $ , the polynomial would be a perfect square. Therefore, adding and subtracting $ 4x{2}y{2} $ yields a difference of two squares, which can be factored:

\[ \begin{aligned}x^{4}+2x^{2}y^{2}+9y^{4}&=x^{4}+6x^{2}y^{2}+9y^{4}-4x^{2}y^{2}\\&=(x^{2}+3y^{2})^{2}-(2xy)^{2}\\&=(x^{2}+3y^{2}-2xy)(x^{2}+3y^{2}+2xy)\\ \end{aligned} \]

SUPPLEMENTARY PROBLEMS

2.13. Find the degree of (a) 8; (b) $ 8x^{7} $ ; (c) $ 5x^{2} - 5x + 5 $ ; (d) $ 5^{2} - 5+ 5 $ ; (e) $ x^{2} + 2xy + y^{2} - 6x + 8y + 25 $

Ans. (a) 0; (b) 7; (c) 2; (d) 0; (e) 2

2.14. Let P be a polynomial of degree m and Q be a polynomial of degree n. Show that (a) PQ is a polynomial of degree $ m + n $ ; (b) the degree of $ P + Q $ is less than or equal to the larger of m, n.

2.15. Let $ A = x^{2} - xy + 2y^{2} $ , $ B = x^{3} - y^{3} $ , $ C = 2x^{2} - 5x + 4 $ , $ D = 3x^{2} - 2y^{2} $ . Find

  1. $ A + D $ ; (b) BD; (c) B - Cx; (d) $ x{2}A{2} - B^{2} $ ; (e) $ AD - B^{2} $

Ans. (a) $ 4x^{2} - xy $ ; (b) $ 3x^{5} - 2x{3}y{2} - 3x{2}y{3} + 2y^{5} $ ; (c) $ -x^{3} - y^{3} + 5x^{2} - 4x $ ;

\[ \left(\mathrm{d}\right)\ -2x^{5}y+5x^{4}y^{2}-2x^{3}y^{3}+4x^{2}y^{4}-y^{6}; \]

\[ \left(\mathrm{e}\right)3x^{4}-3x^{3}y+4x^{2}y^{2}+2xy^{3}-4y^{4}-x^{6}+2x^{3}y^{3}-y^{6} \]

2.16. Using the definitions of the previous problem, subtract C from the sum of A and D.

Ans. $ 2x^{2}-xy+5x-4 $

2.17. Perform indicated operations: (a) $ -(x-5)^{2} $ ; (b) $ 2x-(x-3)^{2} $ ; (c) $ 5a(2a-1){2}-3(a-2){3} $ ;

  1. $ -(4x+1){3}-2(4x+1){2} $

Ans. (a) $ -x^{2} + 10x - 25 $ ; (b) $ -x^{2} + 8x - 9 $ ; (c) $ 17a^{3} - 2a^{2} - 31a + 24 $ ;

\[ \left(d\right)-64x^{3}-80x^{2}-28x-3 \]

2.18. Perform indicated operations: (a) $ -3(x-2)^{2} $ ; (b) $ -3-4(x+4)^{2} $ ; (c) $ 4(x+3){2}-3(x-2){2} $ ;

  1. $ (x+3)(x+4)-(x+5)^{2} $ ; (e) $ -(x+2){3}-(x+2){2}-5(x+2)+10 $

Ans. (a) $ -3x^{2} + 12x - 12 $ ; (b) $ -4x^{2} - 32x - 67 $ ; (c) $ x^{2} + 36x + 24 $ ;

\[ \left(\mathrm{d}\right)-3x-13;\left(\mathrm{e}\right)-x^{3}-7x^{2}-21x-12 \]

2.19. Perform indicated operations: (a) $ (x - h)^{2} + (y - k)^{2} $ ; (b) $ (x + h)^{4} - x^{4} $ ;

  1. $ R^{2} - (R - x)^{2} $ ; (d) $ (ax + by + c)^{2} $

\[ Ans.\quad(a)x^{2}-2xh+h^{2}+y^{2}-2yk+k^{2};(b)4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}; \]

\[ \mathrm{(c)~}2R x-x^{2};\mathrm{(d)~}a^{2}x^{2}+b^{2}y^{2}+c^{2}+2a b x y+2a c x+2b c y \]

2.20. Factor: (a) $ x^{2}-12x+27 $ ; (b) $ x^{2}+10x+25 $ ; (c) $ x{4}-6x{2}+9 $ ; (d) $ x^{3}-64 $ ;

Ans. (a) $ (x-3)(x-9) $ ; (b) $ (x+5)^{2} $ ; (c) $ (x{2}-3){2} $ ; (d) $ (x-4)(x^{2}+4x+16) $ ;

\[ \begin{array}{l}(e)(3x-10)(x+1);(f)3x(x+6)(x-1);(g)x^{2}(x+1)(x^{2}-x+1);\end{array} \]

\[ \begin{aligned}(h)\ (x^{2}+2)(2x-3)(2x+3);(i)\ (x^{2}-3xy-y^{2})(x^{2}+3xy-y^{2})\end{aligned} \]

2.21. Factor: (a) $ t^{2} + 6t - 27 $ ; (b) $ 4x^{3} - 20x^{2} - 24x $ ; (c) $ 3x^{2} - x - 14 $ ; (d) $ 5x^{2} - 3x - 14 $ ; (e) $ 4x^{6} - 37x^{3} + 9 $ ;

\[ \mathrm{f)}\ (x-2)^{3}-(x-2)^{2};\ (\mathrm{g})\ x^{2}-6x+9-y^{2}-2yz-z^{2};\ (\mathrm{h})\ 16x^{4}-x^{2}y^{2}+y^{4} \]

Ans. (a) $ (t + 9)(t - 3) $ ; (b) $ 4x(x + 1)(x - 6) $ ; (c) $ (3x - 7)(x + 2) $ ; (d) $ (5x + 7)(x - 2) $ ;

\[ \begin{array}{l}(e)(4x^{3}-1)(x^{3}-9);(f)(x-2)^{2}(x-3);(g)(x-3-y-z)(x-3+y+z);\end{array} \]

\[ \left(\mathrm{h}\right)(4x^{2}+y^{2}-3xy)(4x^{2}+y^{2}+3xy) \]

2.22. Factor: (a) $ x^{2} - 6xy + 9y^{2} $ ; (b) $ x^{4} - 5x^{2} + 4 $ ; (c) $ x^{4} - 3x^{2} - 4 $ ; (d) $ x^{3} + y^{3} + x^{2} - y^{2} $ ;

Ans. (a) $ (x - 3y)^{2} $ ; (b) $ (x - 1)(x + 1)(x - 2)(x + 2) $ ; (c) $ (x - 2)(x + 2)(x^{2} + 1) $ ;

  1. $ (x + y)(x^{2} - xy + y^{2} + x - y) $ ; (e) $ P(1 + r)^{3} $ ;

\[ \begin{aligned}(f)(ax-2y)(ax+2y)(a^{2}x^{2}+2axy+4y^{2})(a^{2}x^{2}-2axy+4y^{2});\end{aligned} \]

\[ (g)(a^{2}x^{2}+4y^{2})(a^{4}x^{4}-4a^{2}x^{2}y^{2}+16y^{4}) \]

2.23. Factor: (a) $ x{5}(x+2){3}+x{4}(x+2){4} $ ; (b) $ 5x{4}(3x-5){4}+12x{5}(3x-5){3} $ ;

  1. $ 2(x+3)(x+5){4}+4(x+3){2}(x+5)^{3} $ ; (d) $ 3(5x+2){2}(5)(3x-4){4}+(5x+2){3}(4)(3x-4){3} $ ;

  2. $ 5(x^{2} + 4)^{4}(8x - 1)^{2}(2x) + 2(x^{2} + 4)^{5}(8x - 1)(8) $

Ans. (a) $ x{4}(x+2){3}(2x+2) $ ; (b) $ x{4}(3x-5){3}(27x-25) $ ; (c) $ 2(x+3)(x+5)^{3}(3x+11) $ ;

\[ \begin{aligned}(d)\ 3(5x+2)^{2}(3x-4)^{3}(35x-12);(e)\ 2(x^{2}+4)^{4}(8x-1)(48x^{2}-5x+32)\end{aligned} \]