Using the factor theorem, show that 2x + 1 is a factor of 2x^3 + 3x^2 – 11x – 6.
The Correct Answer and Explanation is:
To show that 2x+12x + 12x+1 is a factor of 2×3+3×2−11x−62x^3 + 3x^2 – 11x – 62×3+3×2−11x−6 using the Factor Theorem, follow these steps:
Step 1: Use the Factor Theorem
The Factor Theorem states:
If f(a)=0f(a) = 0f(a)=0, then (x−a)(x – a)(x−a) is a factor of the polynomial f(x)f(x)f(x).
Given the expression 2x+12x + 12x+1, rewrite it in the form x−ax – ax−a.
To do this, solve: 2x+1=0⇒x=−122x + 1 = 0 \Rightarrow x = -\frac{1}{2}2x+1=0⇒x=−21
Now evaluate the polynomial: f(x)=2×3+3×2−11x−6f(x) = 2x^3 + 3x^2 – 11x – 6f(x)=2×3+3×2−11x−6
at x=−12x = -\frac{1}{2}x=−21.
Step 2: Substitute x=−12x = -\frac{1}{2}x=−21 into the polynomial
f(−12)=2(−12)3+3(−12)2−11(−12)−6f\left(-\frac{1}{2}\right) = 2\left(-\frac{1}{2}\right)^3 + 3\left(-\frac{1}{2}\right)^2 – 11\left(-\frac{1}{2}\right) – 6f(−21)=2(−21)3+3(−21)2−11(−21)−6
Now calculate each term:
- (−12)3=−18\left(-\frac{1}{2}\right)^3 = -\frac{1}{8}(−21)3=−81
- (−12)2=14\left(-\frac{1}{2}\right)^2 = \frac{1}{4}(−21)2=41
f(−12)=2(−18)+3(14)+112−6f\left(-\frac{1}{2}\right) = 2\left(-\frac{1}{8}\right) + 3\left(\frac{1}{4}\right) + \frac{11}{2} – 6f(−21)=2(−81)+3(41)+211−6 =−28+34+112−6=−14+34+112−6= -\frac{2}{8} + \frac{3}{4} + \frac{11}{2} – 6 = -\frac{1}{4} + \frac{3}{4} + \frac{11}{2} – 6=−82+43+211−6=−41+43+211−6
Combine the terms:
- −14+34=24=12-\frac{1}{4} + \frac{3}{4} = \frac{2}{4} = \frac{1}{2}−41+43=42=21
- 12+112=6\frac{1}{2} + \frac{11}{2} = 621+211=6
- 6−6=06 – 6 = 06−6=0
f(−12)=0f\left(-\frac{1}{2}\right) = 0f(−21)=0
Step 3: Conclusion
Since f(−12)=0f\left(-\frac{1}{2}\right) = 0f(−21)=0, by the Factor Theorem, x+12x + \frac{1}{2}x+21 is a factor of f(x)f(x)f(x).
Multiplying both sides of x+12x + \frac{1}{2}x+21 by 2 gives 2x+12x + 12x+1, so: 2x+1 is a factor of 2×3+3×2−11x−6\boxed{2x + 1 \text{ is a factor of } 2x^3 + 3x^2 – 11x – 6}2x+1 is a factor of 2×3+3×2−11x−6
Summary Explanation (Like in textbooks)
To determine whether 2x+12x + 12x+1 is a factor of the polynomial f(x)=2×3+3×2−11x−6f(x) = 2x^3 + 3x^2 – 11x – 6f(x)=2×3+3×2−11x−6, the Factor Theorem is applied. First, rewrite 2x+12x + 12x+1 in the form x−ax – ax−a by solving 2x+1=02x + 1 = 02x+1=0, which gives x=−12x = -\frac{1}{2}x=−21. Then evaluate the polynomial at x=−12x = -\frac{1}{2}x=−21. If the result is 0, then 2x+12x + 12x+1 is indeed a factor. Substituting x=−12x = -\frac{1}{2}x=−21 into the polynomial results in zero, confirming that the factor condition is satisfied. Thus, 2x+12x + 12x+1 is a factor of the polynomial.
