Determine the linear and quadratic factors of the expression 2x4+19x3+35x2.(1 point)%0D%0AResponses%0D%0A%0D%0Ax2(2x+5)(x+7)%0D%0Ax squared left parenthesis 2 x plus 5 right parenthesis left parenthesis x plus 7 right parenthesis%0D%0A%0D%0A(2x+5)(x+7)%0D%0Aleft parenthesis 2 x plus 5 right parenthesis left parenthesis x plus 7 right parenthesis%0D%0A%0D%0A(2x3+5x2)(x+7)%0D%0Aleft parenthesis 2 x cubed plus 5 x squared right parenthesis left parenthesis x plus 7 right parenthesis%0D%0A%0D%0A(2x+5)(x3+7x2)

Answer 1

To determine the linear and quadratic factors of the expression \(2x^4 + 19x^3 + 35x^2\), we can start by factoring out the greatest common factor (GCF) from the polynomial.

1. Identify the GCF: The GCF of the terms \(2x^4\), \(19x^3\), and \(35x^2\) is \(x^2\).

2. Factor out the GCF: Factor \(x^2\) from the polynomial: \[ 2x^4 + 19x^3 + 35x^2 = x^2(2x^2 + 19x + 35) \]

3. Factor the quadratic \(2x^2 + 19x + 35\): We want to factor the quadratic \(2x^2 + 19x + 35\). We can do this by looking for two numbers that multiply to \(2 * 35 = 70\) and add to \(19\). The numbers \(14\) and \(5\) work since \(14 \times 5 = 70\) and \(14 + 5 = 19\).

   Now, we rewrite the middle term using these numbers: \[ 2x^2 + 14x + 5x + 35 \] Group the terms: \[ = 2x(x + 7) + 5(x + 7) \] We can factor out \((x + 7)\): \[ = (2x + 5)(x + 7) \]

4. Combine all factors: Putting it all together, we have: \[ 2x^4 + 19x^3 + 35x^2 = x^2(2x + 5)(x + 7) \]

Thus, the complete factorization gives us the linear factors \(2x + 5\) and \(x + 7\), and the quadratic factor \(x^2\).

The answer to your question is: \(x^2(2x + 5)(x + 7)\).