Factor this polynomial completely.
[tex]x^2+3 x-18[/tex]
A. [tex](x-2)(x+9)[/tex]
B. [tex](x-3)(x+6)[/tex]
C. [tex](x-3)(x-6)[/tex]
D. [tex](x+3)(x-6)[/tex]
Answer (1)
Find two numbers that multiply to -18 and add to 3.
The numbers are -3 and 6.
Express the polynomial as a product of two binomials using the two numbers: ( x − 3 ) ( x + 6 ) .
The factored form of the polynomial is ( x − 3 ) ( x + 6 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic polynomial x 2 + 3 x − 18 . This means we want to find two binomials of the form ( x + a ) ( x + b ) such that their product equals the given polynomial.
Finding the Correct Factors To factor the polynomial x 2 + 3 x − 18 , we need to find two numbers that multiply to -18 and add up to 3. Let's list the factor pairs of 18: (1, 18), (2, 9), and (3, 6). Since the product is -18, one of the numbers in each pair must be negative. We need to find a pair where the sum is 3.
Determining the Signs Consider the pair (3, 6). If we make 3 negative, we have -3 and 6. Their product is − 3 × 6 = − 18 , and their sum is − 3 + 6 = 3 . This is exactly what we need!
Writing the Factored Form Therefore, we can write the polynomial as ( x − 3 ) ( x + 6 ) .
Final Answer Thus, the factored form of the polynomial x 2 + 3 x − 18 is ( x − 3 ) ( x + 6 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Imagine you are designing a rectangular garden with an area represented by the expression x 2 + 3 x − 18 . By factoring this expression into ( x − 3 ) ( x + 6 ) , you determine the dimensions of the garden, ensuring it meets specific area requirements and fits within the available space. This skill is also crucial in physics for solving projectile motion problems, where factoring helps determine the time it takes for an object to reach a certain height.
