Determine the factors of the polynomial equation y=2x^3-7x^2-8x+28 - square 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 />1. **Find a root of the polynomial:**<br /> We need to find a value of \( x \) that makes the polynomial equal to zero. Let's test some possible rational roots using the Rational Root Theorem, which states that any rational root of the polynomial \( 2x^3 - 7x^2 - 8x + 28 \) must be a factor of the constant term (28) divided by a factor of the leading coefficient (2). Possible rational roots are \( \pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{2}, \pm \frac{7}{2} \).<br /><br /> Testing \( x = 2 \):<br /> \[<br /> y = 2(2)^3 - 7(2)^2 - 8(2) + 28 = 2(8) - 7(4) - 8(2) + 28 = 16 - 28 - 16 + 28 = 0<br /> \]<br /> So, \( x = 2 \) is a root.<br /><br />2. **Use synthetic division to factor out \( x - 2 \):**<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 />3. **Factor the quadratic polynomial \( 2x^2 - 3x - 14 \):**<br /><br /> To factor \( 2x^2 - 3x - 14 \), we look for two numbers that multiply to \( 2 \cdot (-14) = -28 \) 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 />4. **Combine all factors:**<br /><br /> The complete factorization of the polynomial \( 2x^3 - 7x^2 - 8x + 28 \) is:<br /> \[<br /> y = 2x^3 - 7x^2 - 8x + 28 = (x - 2)(2x - 7)(x + 2)<br /> \]<br /><br />5. **Set each factor equal to zero to solve the polynomial equation:**<br /><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 /> \[<br /> (x + 2) = 0 \quad \Rightarrow \quad x = -2<br /> \]<br /><br />So, the factors of the polynomial equation \( y = 2x^3 - 7x^2 - 8x + 28 \) are \( (x - 2)(2x - 7)(x + 2) \), and the solutions to the equation are \( x = 2, \frac{7}{2}, \) and \( -2 \).