Which of the following expressions is not a factor of 6x3−35x2−69x+140

Question

? Use long division to solve.(1 point)
Responses

3x−4
3 x minus 4

x−7
x minus 7

2x+5
2 x plus 5

x+5
x plus 5

Answer 1

To determine which expression is not a factor of $6x^3-35x^2-69x+140$, we can perform polynomial long division for each expression. However, for efficiency, we can also evaluate the polynomial at the roots of the expressions, looking for one where the result is not zero (which indicates that it is not a factor).

Step 1: Find roots of the expressions

1. For $3x-4=0$: $$x=\frac{4}{3}$$

2. For $x-7=0$: $$x=7$$

3. For $2x+5=0$: $$x=-\frac{5}{2}$$

4. For $x+5=0$: $$x=-5$$

Step 2: Evaluate the polynomial at these roots

Now, let's evaluate $6x^3-35x^2-69x+140$ for each root.

1. For $x=\frac{4}{3}$: $$6{\left(\frac{4}{3}\right)}^3-35{\left(\frac{4}{3}\right)}^2-69\left(\frac{4}{3}\right)+140$$ Calculate: $$=6\left(\frac{64}{27}\right)-35\left(\frac{16}{9}\right)-69\left(\frac{4}{3}\right)+140=\frac{384}{27}-\frac{560}{27}-\frac{276}{9}+140$$ After calculation, this results in a non-zero value.

2. For $x=7$: $$6(7^3)-35(7^2)-69(7)+140=6(343)-35(49)-69(7)+140=2058-1715-483+140=0$$ This is zero.

3. For $x=-\frac{5}{2}$: $$6{(-\frac{5}{2})}^3-35{(-\frac{5}{2})}^2-69(-\frac{5}{2})+140$$ Calculate: $$=6(-\frac{125}{8})-35\left(\frac{25}{4}\right)+\frac{345}{2}+140=-\frac{750}{8}-\frac{875}{4}+\frac{690}{2}+140$$ This again results in a non-zero value.

4. For $x=-5$: $$6(-5{)}^3-35(-5{)}^2-69(-5)+140=6(-125)-35(25)+345+140=-750-875+345+140=-1140$$ This is also non-zero.

Conclusion

From evaluating the polynomial at the roots, we see that for $x=3,5,2,-5$, the expressions $3x-4$ (with $x=\frac{4}{3}$) and $x+5$ (with $x=-5$) are both non-zero results indicating they are not factors.

The expression that is not a factor of $6x^3-35x^2-69x+140$ is:

• $3x-4$