Question

1. factor the following polynomial completely. if it cannot be factored, write prime polynomial.

10x² + 13x - 9
*5. factor the following polynomial completely. if it cannot be factored, write prime polynomial.
8x² + 18x + 9
*6. factor the following polynomial completely. if it cannot be factored, write prime polynomial.
5m⁵n + 14m⁴n² + 8m³n²

Explanation:

Problem 4: Factor \(10x^2 + 13x - 9\)

Step 1: Find two numbers

We need two numbers \(a\) and \(b\) such that \(a\times b=10\times(-9) = - 90\) and \(a + b=13\). After checking, we find that \(a = 18\) and \(b=- 5\) since \(18\times(-5)=-90\) and \(18+( - 5)=13\).

Step 2: Rewrite the middle term

Rewrite the middle term \(13x\) as \(18x-5x\), so the polynomial becomes \(10x^{2}+18x - 5x-9\).

Step 3: Group and factor

Group the first two terms and the last two terms: \((10x^{2}+18x)-(5x + 9)\). Factor out the greatest common factor from each group: \(2x(5x + 9)-1(5x + 9)\).

Step 4: Factor out the common binomial

Factor out \((5x + 9)\) from the two terms: \((2x - 1)(5x+9)\).

Step 1: Find two numbers

We need two numbers \(a\) and \(b\) such that \(a\times b=8\times9 = 72\) and \(a + b = 18\). We find that \(a=12\) and \(b = 6\) since \(12\times6=72\) and \(12 + 6=18\).

Step 2: Rewrite the middle term

Rewrite the middle term \(18x\) as \(12x + 6x\), so the polynomial becomes \(8x^{2}+12x+6x + 9\).

Step 3: Group and factor

Group the first two terms and the last two terms: \((8x^{2}+12x)+(6x + 9)\). Factor out the greatest common factor from each group: \(4x(2x + 3)+3(2x + 3)\).

Step 4: Factor out the common binomial

Factor out \((2x + 3)\) from the two terms: \((4x + 3)(2x+3)\).

Step 1: Factor out the GCF

First, factor out the greatest common factor (GCF) of the three terms. The GCF of \(5m^{5}n\), \(14m^{4}n^{2}\) and \(8m^{3}n^{2}\) is \(m^{3}n\). So we have \(m^{3}n(5m^{2}+14mn + 8n^{2})\).

Step 2: Factor the quadratic in \(m\) and \(n\)

Now we factor \(5m^{2}+14mn + 8n^{2}\). We need two numbers \(a\) and \(b\) such that \(a\times b=5\times8=40\) and \(a + b = 14\). We find that \(a = 10\) and \(b = 4\) since \(10\times4 = 40\) and \(10+4=14\). Rewrite the middle term \(14mn\) as \(10mn+4mn\), so \(5m^{2}+10mn+4mn + 8n^{2}\). Group the terms: \((5m^{2}+10mn)+(4mn + 8n^{2})\). Factor out the GCF from each group: \(5m(m + 2n)+4n(m + 2n)\). Factor out \((m + 2n)\): \((5m + 4n)(m + 2n)\).

Step 3: Combine the factors

Putting it all together, the factored form is \(m^{3}n(5m + 4n)(m + 2n)\).

Answer:

\((2x - 1)(5x + 9)\)

Problem 5: Factor \(8x^2+18x + 9\)