Problem 85 - Introductory Algebra for College Students

Question

Explain how to factor \(x^{2}+8 x+15\)

Step-by-Step Solution

Answer

The factor form of \(x^{2} + 8x + 15\) is \((x + 5)(x + 3)\)

Step 1: Find ‘ac’ and ‘b’

The quadratic equation is of the form \( ax^{2} + bx + c \) so \( a=1 \), \( b=8 \), and \( c=15 \). We use ‘ac’ method, so multiply \( a \) and \( c \), which gives \( ac = 1*15 = 15 \). ‘b’ is the coefficient of \( x \), which is 8.

Step 2: Find two numbers

Find two numbers that multiply to \( ac=15 \) and add to \( b=8 \). After some trials, you'll find these two numbers are 5 and 3.

Step 3: Rewrite the expression

Rewrite the middle term (the term with x) of the quadratic equation. Break \(8x\) into \(5x + 3x\) such that the equation becomes \(x^{2} + 5x + 3x + 15\). We split the term \(8x\) into \(5x+3x\), because 5 and 3 are the two numbers found that multiply to 15 and add to 8.

Step 4: Factor by grouping

We can now perform factoring by grouping, which is a method to factorise four-term polynomials. Group the first two terms and the last two terms separately, \(x^{2} + 5x\) and \(3x + 15\). Each group has a common factor, which can be factored out: \(x(x + 5) + 3(x + 5)\).

Step 5: Final factor form

After the above step, you will realize that we can factor out even further since \(x + 5\) is common in both terms. So the final factor form is \((x + 5)(x + 3)\).