To find the third linear factor of the polynomial \( 16x^3 + 68x^2 + 32x - 35 \) given that \( 4x + 5 \) and \( 2x - 1 \) are known factors, we can use polynomial division.
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Multiply the known factors together: \[ (4x + 5)(2x - 1) \] Expanding this: \[ = 4x \cdot 2x + 4x \cdot (-1) + 5 \cdot 2x + 5 \cdot (-1) \] \[ = 8x^2 - 4x + 10x - 5 \] \[ = 8x^2 + 6x - 5 \]
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Now, divide the original polynomial by this product: We need to divide \( 16x^3 + 68x^2 + 32x - 35 \) by \( 8x^2 + 6x - 5 \).
Using polynomial long division:
- Divide the leading term \( 16x^3 \) by the leading term of the divisor \( 8x^2 \) to get \( 2x \).
- Multiply \( 2x \) by the entire divisor \( 8x^2 + 6x - 5 \): \[ 2x(8x^2 + 6x - 5) = 16x^3 + 12x^2 - 10x \]
- Subtract this from the original polynomial: \[ (16x^3 + 68x^2 + 32x - 35) - (16x^3 + 12x^2 - 10x) = (68x^2 - 12x^2) + (32x + 10x) - 35 \] \[ = 56x^2 + 42x - 35 \]
- Now repeat the division process:
- Divide the leading term \( 56x^2 \) by the leading term of the divisor \( 8x^2 \) to get \( 7 \).
- Multiply \( 7 \) by the entire divisor \( 8x^2 + 6x - 5 \): \[ 7(8x^2 + 6x - 5) = 56x^2 + 42x - 35 \]
- Subtract this from what we just calculated: \[ (56x^2 + 42x - 35) - (56x^2 + 42x - 35) = 0 \]
Since we have obtained 0 after performing the division, we conclude that the third factor is: \[ \text{Third factor} = 2x + 7 \]
- Final Result: Thus, the third linear factor of the polynomial \( 16x^3 + 68x^2 + 32x - 35 \) is \( \boxed{2x + 7} \).
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