Which is a factor of $6 x^2+x-7 ?$
A. $x+1$
B. $2 x+1$
C. $6 x+7$
D. $3 x+7$
Answer (1)
Test each option by substituting the value of x that makes the linear expression equal to zero into the quadratic expression.
If the result is zero, then the linear expression is a factor.
Alternatively, factor the quadratic expression 6 x 2 + x − 7 into ( 6 x + 7 ) ( x − 1 ) .
Therefore, 6 x + 7 is a factor of 6 x 2 + x − 7 , so the final answer is 6 x + 7 .
Explanation
Understanding the Problem We are given the quadratic expression 6 x 2 + x − 7 and asked to find which of the given linear expressions is a factor. A factor of a polynomial is an expression that divides the polynomial evenly, leaving no remainder. We can test each option by substituting the value of x that makes the linear expression equal to zero into the quadratic expression. If the result is zero, then the linear expression is a factor. Alternatively, we can factor the quadratic expression and see if any of the given options match one of the factors.
Testing Each Option Let's test each option:
x + 1 : If x + 1 is a factor, then x = − 1 should be a root of the quadratic. Substituting x = − 1 into the quadratic gives 6 ( − 1 ) 2 + ( − 1 ) − 7 = 6 − 1 − 7 = − 2 . Since this is not zero, x + 1 is not a factor.
2 x + 1 : If 2 x + 1 is a factor, then x = − 2 1 should be a root of the quadratic. Substituting x = − 2 1 into the quadratic gives 6 ( − 2 1 ) 2 + ( − 2 1 ) − 7 = 6 ( 4 1 ) − 2 1 − 7 = 2 3 − 2 1 − 7 = 1 − 7 = − 6 . Since this is not zero, 2 x + 1 is not a factor.
6 x + 7 : If 6 x + 7 is a factor, then x = − 6 7 should be a root of the quadratic. Substituting x = − 6 7 into the quadratic gives 6 ( − 6 7 ) 2 + ( − 6 7 ) − 7 = 6 ( 36 49 ) − 6 7 − 7 = 6 49 − 6 7 − 6 42 = 6 49 − 7 − 42 = 6 0 = 0 . Since this is zero, 6 x + 7 is a factor.
3 x + 7 : If 3 x + 7 is a factor, then x = − 3 7 should be a root of the quadratic. Substituting x = − 3 7 into the quadratic gives 6 ( − 3 7 ) 2 + ( − 3 7 ) − 7 = 6 ( 9 49 ) − 3 7 − 7 = 9 294 − 9 21 − 9 63 = 9 294 − 21 − 63 = 9 210 = 3 70 . Since this is not zero, 3 x + 7 is not a factor.
Factoring the Quadratic Expression Alternatively, we can factor the quadratic expression 6 x 2 + x − 7 . We are looking for two numbers that multiply to 6 ( − 7 ) = − 42 and add to 1 . These numbers are 7 and − 6 . So we can rewrite the quadratic as:
6 x 2 − 6 x + 7 x − 7 = 6 x ( x − 1 ) + 7 ( x − 1 ) = ( 6 x + 7 ) ( x − 1 )
Therefore, 6 x + 7 is a factor.
Final Answer Therefore, the factor of 6 x 2 + x − 7 among the given options is 6 x + 7 .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Imagine you're designing a bridge, and you need to ensure that the forces acting on it are balanced. By expressing these forces as a quadratic equation and factoring it, you can determine the critical points where the forces might cause the structure to fail. This helps you design a safer and more stable bridge.
