Question

Factor each expression completely

(a)$4x^2-25$

(b)$2x^2+5x-12$

(c)$x^3-3x^2-4x+12$

(d)$x^4+27x$

(e)$3x^{\frac{3}{2}}-9x^{\frac{1}{2}}+6x^{-\frac{1}{2}}$

(f)$x^{3}y-4xy$

Short answer

(a)$(2x-5)(2x+5)$

(b)$(2x-3)(x+4)$

(c)$(x-2)(x+2)(x-3)$

(d)$x(x+3)(x^2-3x+9)$

(e)$3x^{-\frac{1}{2}}(x-1)(x-2)$

(f)$xy(x-2)(x+2)$

Step-by-step solution

Part a. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1\mathrm{.........},a_n$ are real numbers and n is a non-negative integers.

Part a. Step 2. Given expression.

The expression is$4x^2-25$

Part a. Step 3. Factoring the expression.

$$\begin{gathered} 4x^2-25 \\ ={\left(2x\right)}^2-{\left(5\right)}^2 \\ =(2x-5)(2x+5)[A^2-B^2=(A-B\left)\right(A+B\left)\right] \end{gathered}$$

Part b. Step 1. Concept of factoring trinomials.

To factor a trinomial of the form $x^2+bx+c$,

$(x+r)(x+s)=x^2+(r+s)x+rs$

So, we need to choose number $r\mathrm{and}s$so that$r+s=b\mathrm{and}rs=c$

Part b. Step 2. Given expression.

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

Part b. Step 3. Factoring the expression.

$$\begin{gathered} 2x^2+5x-12 \\ =2x^2+8x-3x-12 \\ =2x(x+4)-3(x+4) \\ =(2x-3)(x+4) \end{gathered}$$

Part c. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1\mathrm{.........},a_n$ are real numbers and n is a non-negative integers.

Part c. Step 2. Given expression.

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

Part c. Step 3. Factoring the expression.

$$\begin{gathered} x^3-3x^2-4x+12 \\ =(x^3-3x^2)-(4x-12)(\mathrm{Group}\mathrm{term}) \\ ={\mathrm{x}}^2(\mathrm{x}-3)-4(\mathrm{x}-3)(\mathrm{Factor}\mathrm{common}\mathrm{factors}) \\ =({\mathrm{x}}^2-4)(\mathrm{x}-3)(\mathrm{Factor}(\mathrm{x}-3)\mathrm{from}\mathrm{each}\mathrm{term}) \\ =(\mathrm{x}-2)(\mathrm{x}+2)(\mathrm{x}-3)(\mathrm{Factor}\mathrm{completely}) \end{gathered}$$

Part d. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1\mathrm{.........},a_n$ are real numbers and n is a non-negative integers.

Part d. Step 2. Given expression.

The expression is$x^4+27x$

Part d. Step 3. Factoring the expression.

$$\begin{gathered} x^4+27x \\ =x(x^3+27)(\mathrm{Common}\mathrm{factor}\mathrm{is}\mathrm{x}) \\ =\mathrm{x}\left[\right(\mathrm{x}+3\left)\right({\mathrm{x}}^2-3\mathrm{x}+9\left)\right] \\ =\mathrm{x}(\mathrm{x}+3)({\mathrm{x}}^2-3\mathrm{x}+9) \end{gathered}$$

Part e. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1\mathrm{.........},a_n$ are real numbers and n is a non-negative integers.

Part e. Step 2. Given expression.

The expression is$3x^{\frac{3}{2}}-9x^{\frac{1}{2}}+6x^{-\frac{1}{2}}$

Part e. Step 3. Factoring the expression.

Factor out the power of x with the smallest exponent, that is $x^{-\frac{1}{2}}$

$$\begin{gathered} 3x^{\frac{3}{2}}-9x^{\frac{1}{2}}+6x^{-\frac{1}{2}} \\ =3x^{-\frac{1}{2}}(x^2-3x+2)(\mathrm{Factor}\mathrm{out}3x^{-\frac{1}{2}}) \\ =3x^{-\frac{1}{2}}(x-1)(x-2)(\mathrm{Factor}\mathrm{the}\mathrm{quadratic}{x}^2-3x+2) \end{gathered}$$

Part f. Step 1. Definition of polynomial.

A polynomial in the variable x is an expression of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.......}+a_{1}x+a_0$where$a_0,a_1\mathrm{.........},a_n$ are real numbers and n is a non negative integers.

Part f. Step 2. Given expression.

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

Part f. Step 3. Factoring the expression.

$$\begin{gathered} x^{3}y-4xy \\ =xy(x^2-4)(\mathrm{Common}\mathrm{factor}\mathrm{is}\mathrm{xy}) \\ =xy(x-2)(x+2)(\mathrm{Factor}{x}^2-4) \end{gathered}$$