Questions: 10. 5x^4 + 20x^3 - 105x^2 8. x^4 + 8x^2 + 15 8. x^4 + 8x^2 + 15 9. x^4 - 16 a = x^2 t = 4 (x^2 + 4)(x^2 - 4) a: x - 6 = 2 (x^2 + 4)(x + 2)(x - 2) 5x^2(x^2 + 4x - 21) 11. (5x^3 - 4x - 1) ÷ (x + 1) 12. (5x^4 - 9x^3 + 8x^2 - 26x - 20) / (x^2 - 3)

10. 5x^4 + 20x^3 - 105x^2 8. x^4 + 8x^2 + 15 8. x^4 + 8x^2 + 15 9. x^4 - 16 a = x^2 t = 4 (x^2 + 4)(x^2 - 4) a: x - 6 = 2 (x^2 + 4)(x + 2)(x - 2) 5x^2(x^2 + 4x - 21) 11. (5x^3 - 4x - 1) ÷ (x + 1) 12. (5x^4 - 9x^3 + 8x^2 - 26x - 20) / (x^2 - 3)
Transcript text: 10. $5 x^{4}+20 x^{3}-105 x^{2}$ 8. $x^{4}+8 x^{2}+15$ 8. $x^{4}+8 x^{2}+15$ 9. $x^{4}-16$ $a=x^{2} t=4$ $\left(x^{2}+4\right)\left(x^{2}-4\right)$ $a: x-6=2$ $\left(x^{2}+4\right)(x+2)(x-2)$ $5 x^{2}\left(x^{2}+4 x-21\right)$ 11. $\left(5 x^{3}-4 x-1\right) \div(x+1)$ 12. $\frac{5 x^{4}-9 x^{3}+8 x^{2}-26 x-20}{x^{2}-3}$.
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Solution

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Solution Steps

Solution Approach

  1. For the expression \(5x^4 + 20x^3 - 105x^2\), factor out the greatest common factor (GCF) from the polynomial.

  2. For the polynomial \(x^4 + 8x^2 + 15\), factor it by recognizing it as a quadratic in terms of \(x^2\).

  3. For the division \((5x^3 - 4x - 1) \div (x + 1)\), use polynomial long division to find the quotient.

Step 1: Factor \(5x^4 + 20x^3 - 105x^2\)

To factor the polynomial \(5x^4 + 20x^3 - 105x^2\), we first identify the greatest common factor (GCF), which is \(5x^2\). Factoring this out, we have: \[ 5x^4 + 20x^3 - 105x^2 = 5x^2(x - 3)(x + 7) \]

Step 2: Factor \(x^4 + 8x^2 + 15\)

Next, we factor the polynomial \(x^4 + 8x^2 + 15\). We can treat this as a quadratic in terms of \(x^2\): \[ x^4 + 8x^2 + 15 = (x^2 + 3)(x^2 + 5) \]

Step 3: Divide \((5x^3 - 4x - 1)\) by \((x + 1)\)

For the division of the polynomial \((5x^3 - 4x - 1)\) by \((x + 1)\), we perform polynomial long division, yielding: \[ \frac{5x^3 - 4x - 1}{x + 1} = 5x^2 - 5x + 1 \quad \text{with a remainder of } -2 \]

Final Answer

The factored forms and the result of the division are as follows:

  1. \(5x^4 + 20x^3 - 105x^2 = 5x^2(x - 3)(x + 7)\)
  2. \(x^4 + 8x^2 + 15 = (x^2 + 3)(x^2 + 5)\)
  3. \(\frac{5x^3 - 4x - 1}{x + 1} = 5x^2 - 5x + 1\) with a remainder of \(-2\)

Thus, the final boxed answers are: \[ \boxed{5x^4 + 20x^3 - 105x^2 = 5x^2(x - 3)(x + 7)} \] \[ \boxed{x^4 + 8x^2 + 15 = (x^2 + 3)(x^2 + 5)} \] \[ \boxed{\frac{5x^3 - 4x - 1}{x + 1} = 5x^2 - 5x + 1 \text{ with remainder } -2} \]

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