Chapter 1, Problem 7T

(a)

To determine

The factors of the expression $4x^2-25$.

Answer to Problem 7T

The factors of the expression $4x^2-25$ is $\left(2x-5\right)\left(2x+5\right)$.

Explanation of Solution

Given:

The expression is $4x^2-25$.

Calculation:

The formula of difference of squares is,

$A^2-B^2=\left(A-B\right)\left(A+B\right)$ (1)

Rearrange the given expression,

$4x^2-25={\left(2x\right)}^2-{\left(5\right)}^2$

Substitute $2x$ for A and 5 for B in the equation (1) to get factors,

${\left(2x\right)}^2-{\left(5\right)}^2=\left(2x-5\right)\left(2x+5\right)$

Thus, the factors of the expression $4x^2-25$ is $\left(2x-5\right)\left(2x+5\right)$.

(b)

To determine

The factors of the expression $2x^2+5x-12$.

Answer to Problem 7T

The factors of the expression $2x^2+5x-12$ is $\left(2x-3\right)\left(x+4\right)$.

Explanation of Solution

Given:

The expression is $2x^2+5x-12$.

Calculation:

Solve the given expression to get factors,

$2x^2+5x-12=2x^2+8x-3x-12$

The greatest common factor of terms $2x^2$ and $8x$ is $2x$ and the terms $-3x$ and $-12$ is $-3$ to solve the above expression,

$\begin{array}{c}2x^2+5x-12=2x^2+8x-3x-12\\ =2x\left(x+4\right)-3\left(x+4\right)\\ =\left(2x-3\right)\left(x+4\right)\end{array}$

Thus, the factors of the expression $2x^2+5x-12$ is $\left(2x-3\right)\left(x+4\right)$.

(c)

To determine

The factors of the polynomial expression $x^3-3x^2-4x+12$.

Answer to Problem 7T

The factors of the polynomial expression $x^3-3x^2-4x+12$ is $\left(x-3\right)\left(x-2\right)\left(x+2\right)$.

Explanation of Solution

Given:

The polynomial expression is $x^3-3x^2-4x+12$.

Calculation:

First, factor the group terms than factor out common factor in the above expression,

$\begin{array}{c}{x}^3-3x^2-4x+12=x^2\left(x-3\right)-4\left(x-3\right)\\ =\left(x^2-{\left(2\right)}^2\right)\left(x-3\right)\\ =\left(x-2\right)\left(x+2\right)\left(x-3\right)\end{array}$

Thus, the factors of the polynomial expression $x^3-3x^2-4x+12$ is $\left(x-3\right)\left(x-2\right)\left(x+2\right)$.

(d)

To determine

The factors of the expression $x^4+27x$.

Answer to Problem 7T

The factors of the expression $x^4+27x$ is $x\left(x+3\right)\left(x^2-3x+9\right)$.

Explanation of Solution

Given:

The expression is $x^4+27x$.

Calculation:

The greatest common factor of terms $x^4$ and $27x$ is $x$ to solve the above expression,

$x^4+27x=x\left(x^3+{\left(3\right)}^3\right)$

The formula of sum of cubes is,

$A^3+B^3=\left(A+B\right)\left(A^2-AB+B^2\right)$

Substitute x for A and 3 for B in above formula,

$\begin{array}{c}{x}^4+27x=x\left(x^3+{\left(3\right)}^3\right)\\ =x\left\{\left(x+3\right)\left(x^2-x\cdot 3+3^2\right)\right\}\\ =x\left(x+3\right)\left(x^2-3x+9\right)\end{array}$

Thus, the factors of the expression $x^4+27x$ is $x\left(x+3\right)\left(x^2-3x+9\right)$.

(e)

To determine

The factors of the expression $3x^{3/2}-9x^{1/2}+6x^{-1/2}$.

Answer to Problem 7T

The factors of the expression $3x^{3/2}-9x^{1/2}+6x^{-1/2}$ is $3x^{-1/2}\left(x-1\right)\left(x-2\right)$.

Explanation of Solution

Given:

The expression is $3x^{3/2}-9x^{1/2}+6x^{-1/2}$.

Calculation:

First, factor out the power of x with the smallest exponent that is $x^{-1/2}$ than solve the given expression,

$\begin{array}{c}3x^{3/2}-9x^{1/2}+6x^{-1/2}=3x^{-1/2}\left(x^2-3x+2\right)\\ =3x^{-1/2}\left(x^2-2x-x+2\right)\\ =3x^{-1/2}\left\{x\left(x-2\right)-1\left(x-2\right)\right\}\\ =3x^{-1/2}\left(x-1\right)\left(x-2\right)\end{array}$

Thus, the factors of the expression $3x^{3/2}-9x^{1/2}+6x^{-1/2}$ is $3x^{-1/2}\left(x-1\right)\left(x-2\right)$.

(f)

To determine

The factors of the expression $x^{3}y-4xy$.

Answer to Problem 7T

The factors of the expression $x^{3}y-4xy$ is $xy\left(x-2\right)\left(x+2\right)$.

Explanation of Solution

Given:

The expression is $x^{3}y-4xy$.

Calculation:

First, factor out the power of x and y with the smallest exponent that is $xy$ than solve the given expression,

$\begin{array}{c}{x}^{3}y-4xy=xy\left(x^2-{\left(2\right)}^2\right)\\ =xy\left(x-2\right)\left(x+2\right)\end{array}$

Thus, the factors of the expression $x^{3}y-4xy$ is $xy\left(x-2\right)\left(x+2\right)$.