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For the polynomial below, is a zero.Express as a product of linear factors.
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Q. For the polynomial below, is a zero.Express as a product of linear factors.
- Identify Zero and Factor: Since is a zero of the polynomial , we know that is a factor of . To find the other factors, we can perform polynomial division or use synthetic division to divide by .
- Perform Synthetic Division: Let's use synthetic division to divide by . We set up the synthetic division as follows:\begin{array}{r|rrrr} 1 & 1 & -7 & 12 & -6 \ & \underline{+1} & \underline{-6} & \underline{6} \ & 1 & -6 & 6 & 0 \end{array}The remainder is , which confirms that is indeed a zero of . The quotient from the synthetic division is .
- Factor Quadratic Polynomial: Now we need to factor the quadratic polynomial . We look for two numbers that multiply to and add up to . The numbers and satisfy these conditions.
- Express as Linear Factors: We can now express the quadratic polynomial as a product of two linear factors: .
- Express as Linear Factors: We can now express the quadratic polynomial as a product of two linear factors: .Combining the linear factor we found from the zero of the polynomial with the factors of the quadratic, we can express as a product of linear factors: .
