Determine the factors of the polynomial equation y=2x^3-7x^2-8x+28 (a) square (1) PLAY (D) STOP (b) In order to solve the polynomial equation you need to set each factor equal to square

To determine the factors of the polynomial equation \( y = 2x^3 - 7x^2 - 8x + 28 \), we can use the factor theorem and synthetic division.<br /><br />### Step-by-Step Solution:<br /><br />1. **Find Possible Rational Roots:**<br /> According to the Rational Root Theorem, the possible rational roots are the factors of the constant term (28) divided by the factors of the leading coefficient (2). Thus, the possible rational roots are:<br /> \[<br /> \pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{2}<br /> \]<br /><br />2. **Test Possible Roots:**<br /> We test these possible roots using synthetic division or direct substitution into the polynomial to see if they yield zero.<br /><br /> Let's start with \( x = 2 \):<br /> \[<br /> y = 2(2)^3 - 7(2)^2 - 8(2) + 28 = 16 - 28 - 16 + 28 = 0<br /> \]<br /> Since \( x = 2 \) is a root, \( x - 2 \) is a factor.<br /><br />3. **Perform Synthetic Division:**<br /> Divide \( 2x^3 - 7x^2 - 8x + 28 \) by \( x - 2 \):<br /><br /> \[<br /> \begin{array}{r|rrrr}<br /> 2 & 2 & -7 & -8 & 28 \\<br /> & & 4 & -6 & -28 \\<br /> \hline<br /> & 2 & -3 & -14 & 0 \\<br /> \end{array}<br /> \]<br /><br /> The quotient is \( 2x^2 - 3x - 14 \).<br /><br />4. **Factor the Quotient Polynomial:**<br /> Now, we need to factor \( 2x^2 - 3x - 14 \). We look for two numbers that multiply to \( 2 \times (-14) = -28 \) and add up to \( -3 \). These numbers are \( -7 \) and \( 4 \).<br /><br /> Rewrite the middle term:<br /> \[<br /> 2x^2 - 7x + 4x - 14<br /> \]<br /><br /> Factor by grouping:<br /> \[<br /> (2x^2 - 7x) + (4x - 14) = x(2x - 7) + 2(2x - 7) = (2x - 7)(x + 2)<br /> \]<br /><br />5. **Combine All Factors:**<br /> Therefore, the complete factorization of the polynomial is:<br /> \[<br /> y = 2(x - 2)(2x - 7)(x + 2)<br /> \]<br /><br />### Answers:<br />(a) The factors of the polynomial equation are:<br />\[<br />2(x - 2)(2x - 7)(x + 2)<br />\]<br /><br />(b) In order to solve the polynomial equation, you need to set each factor equal to:<br />\[<br />0<br />\]<br /><br />So, the solutions to the polynomial equation \( y = 2x^3 - 7x^2 - 8x + 28 \) are found by solving:<br />\[<br />x - 2 = 0 \quad \Rightarrow \quad x = 2<br />\]<br />\[<br />2x - 7 = 0 \quad \Rightarrow \quad x = \frac{7}{2}<br />\]<br />\[<br />x + 2 = 0 \quad \Rightarrow \quad x = -2<br />\]