QuestionFactor the expression completely. $x^{2}-3 x-18$

Studdy Solution

STEP 1

1. The expression is a quadratic polynomial of the form $ax^2 + bx + c$ .
2. The expression can be factored into the form $(x - p)(x - q)$ , where $p$ and $q$ are the roots of the polynomial.

STEP 2

1. Identify the coefficients $a$ , $b$ , and $c$ in the quadratic polynomial.
2. Find two numbers that multiply to $ac$ and add up to $b$ .
3. Rewrite the middle term using the two numbers found in step 2.
4. Factor by grouping.
5. Write the polynomial as a product of two binomials.

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ in the quadratic polynomial $x^2 - 3x - 18$ .
Here, $a = 1$ , $b = -3$ , and $c = -18$ .

STEP 4

Find two numbers that multiply to $ac$ (which is $1 \cdot -18 = -18$ ) and add up to $b$ (which is $-3$ ).
The numbers that satisfy these conditions are $-6$ and $3$ because: $-6 \cdot 3 = -18$ $-6 + 3 = -3$

STEP 5

Rewrite the middle term $-3x$ using the two numbers found in the previous step.
$x^2 - 3x - 18 = x^2 - 6x + 3x - 18$

STEP 6

Factor by grouping. Group the terms in pairs and factor out the common factors.
$x^2 - 6x + 3x - 18 = x(x - 6) + 3(x - 6)$

STEP 7

Write the polynomial as a product of two binomials. Notice that $(x - 6)$ is a common factor.
$x(x - 6) + 3(x - 6) = (x + 3)(x - 6)$
Thus, the completely factored form of the polynomial $x^2 - 3x - 18$ is:
$(x + 3)(x - 6)$

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