What are the linear and quadratic factors of the expression 3x4+7x3+4x2?(1 point)%0D%0AResponses%0D%0A%0D%0Ax2(3x+4)(x+1)%0D%0Ax squared left parenthesis 3 x plus 4 right parenthesis left parenthesis x plus 1 right parenthesis%0D%0A%0D%0A(3x+4)(x+1)%0D%0Aleft parenthesis 3 x plus 4 right parenthesis left parenthesis x plus 1 right parenthesis%0D%0A%0D%0Ax2(3x+4)(x+1)(x+1)%0D%0Ax squared left parenthesis 3 x plus 4 right parenthesis left parenthesis x plus 1 right parenthesis left parenthesis x plus 1 right parenthesis%0D%0A%0D%0Ax2(3x−4)(x−1)

Question

GPT-4o mini · Answer

To factor the expression $ 3x^4 + 7x^3 + 4x^2 $, we can start by factoring out the greatest common factor (GCF) from the polynomial. 1. The GCF of the terms $ 3x^4 $, $ 7x^3 $, and $ 4x^2 $ is $ x^2 $. 2. Factoring out $ x^2 $ gives: $$ x^2(3x^2 + 7x + 4) $$ Next, we need to factor the quadratic expression $ 3x^2 + 7x + 4 $. We can use the method of factoring by grouping, or look for two numbers that multiply to $ 3 \times 4 = 12 $ and sum to $ 7 $. The numbers $ 3 $ and $ 4 $ work, so we rewrite the middle term: $$ 3x^2 + 3x + 4x + 4 $$ Next, we group the terms: $$ (3x^2 + 3x) + (4x + 4) $$ Factoring each group, we get: $$ 3x(x + 1) + 4(x + 1) $$ Now we factor out the common term $ (x + 1) $: $$ (3x + 4)(x + 1) $$ Putting this all together, we have: $$ x^2(3x + 4)(x + 1) $$ Thus, the linear and quadratic factors of the expression $ 3x^4 + 7x^3 + 4x^2 $ are: $$ \boxed{x^2(3x + 4)(x + 1)} $$