Problem 30 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$ 8 x^{2}+33 x+4 $$

Step-by-Step Solution

Answer

The factored form of \(8 x^{2}+33 x+4\) is \((8x + 1)(x + 4)\).

Step 1: Identify the coefficients and the constant term

The coefficients are 8 for \(x^2\), 33 for \(x\), and the constant term is 4.

Step 2: Find two numbers

Find two numbers that multiply to \(8 \times 4 = 32\) (the product of the coefficient of \(x^2\) and the constant term) and add up to 33 (the coefficient of \(x\)). The numbers that satisfy this condition are 1 and 32.

Step 3: Rewrite the middle term

Rewrite the middle term of the trinomial as the sum of the terms \(33x = x + 32x\).

Step 4: Factor by grouping

Rearrange the expression \(8 x^{2}+x+32 x+4\) and group the first two terms together and the last two terms together to get \(8 x^{2}+x+32x+4 = (8 x^{2}+ x) + (32x + 4)\). Factor each group getting, \(x(8x + 1) + 4(8x + 1)\).

Step 5: Factor out the common binomial

Now take out the common binomial to get the final factored form: \((8x + 1)(x + 4)\).