Math

QuestionFactor the polynomial t4+5t314t2t^{4}+5 t^{3}-14 t^{2} as tN(t+A)(t+B)t^{N}(t+A)(t+B) with A<BA<B. Find NN, AA, and BB.

Studdy Solution

STEP 1

Assumptions1. The given polynomial is t4+5t314tt^{4}+5 t^{3}-14 t^{} . The polynomial can be factored in the form t(t+A)(t+B)t^{}(t+A)(t+B) where A<BA<B

STEP 2

First, we need to factor out the greatest common factor (GCF) from the polynomial. The GCF is the highest power of tt that divides all terms of the polynomial.
t4+5t14t2=t2(t2+5t14)t^{4}+5 t^{}-14 t^{2} = t^{2}(t^{2}+5t-14)

STEP 3

Now, we need to factor the quadratic polynomial t2+5t14t^{2}+5t-14. This can be done by finding two numbers that multiply to 14-14 and add to 55.

STEP 4

The numbers that satisfy these conditions are 77 and 2-2. So, the factored form of the quadratic polynomial is (t2)(t+7)(t-2)(t+7).

STEP 5

Substitute the factored form of the quadratic polynomial back into the equation.
t2(t2+5t14)=t2(t2)(t+7)t^{2}(t^{2}+5t-14) = t^{2}(t-2)(t+7)So, =2=2, A=2A=-2, and B=7B=7.

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