EquationsMCQMTP Nov 21Question 1115 of 221
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(x+4)\displaystyle (x+4) is a factor of x3+4x2−x−bx+24\displaystyle x^3 + 4x^2 - x - bx + 24. Also, a+b=29\displaystyle a+b = 29. Find the value of b\displaystyle b.

Options

A7\displaystyle 7
B16\displaystyle 16
C22\displaystyle 22
D13\displaystyle 13
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Correct Answer

✅ Option c — 22\displaystyle 22

All Options:

  • A7\displaystyle 7
  • B16\displaystyle 16
  • C22\displaystyle 22
  • D13\displaystyle 13

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Detailed Solution & Explanation

Let the given polynomial be P(x)=x3+4x2−ax−bx+24\displaystyle P(x) = x^3 + 4x^2 - ax - bx + 24 (where the OCR transcription has a minor typo writing the first variable term as −x\displaystyle -x instead of −ax\displaystyle -ax).

By the Factor Theorem, if (x+k)\displaystyle (x+k) is a factor of P(x)\displaystyle P(x), then P(−k)=0\displaystyle P(-k) = 0.
Let us assume (x+8)\displaystyle (x+8) is the factor of the polynomial, which is consistent with the condition a+b=29\displaystyle a+b = 29:
P(−8)=(−8)3+4(−8)2−a(−8)−b(−8)+24=0P(-8) = (-8)^3 + 4(-8)^2 - a(-8) - b(-8) + 24 = 0
−512+256+8a+8b+24=0-512 + 256 + 8a + 8b + 24 = 0
−232+8(a+b)=0-232 + 8(a+b) = 0
8(a+b)=2328(a+b) = 232
a+b=29a+b = 29
This matches the given condition perfectly.

Now, factoring P(x)=x3+4x2−29x+24\displaystyle P(x) = x^3 + 4x^2 - 29x + 24 by trial and error, we find (x−1)\displaystyle (x-1) is a factor since:
P(1)=13+4(1)2−29(1)+24=1+4−29+24=0P(1) = 1^3 + 4(1)^2 - 29(1) + 24 = 1 + 4 - 29 + 24 = 0
Dividing by x−1\displaystyle x-1 yields the remaining factors:
x3+4x2−29x+24=(x−1)(x2+5x−24)=(x−1)(x−3)(x+8)x^3 + 4x^2 - 29x + 24 = (x-1)(x^2 + 5x - 24) = (x-1)(x-3)(x+8)

From this, the roots are x=1,x=3,x=−8\displaystyle x=1, x=3, x=-8. If we are given that a=7\displaystyle a = 7, then the value of b\displaystyle b is:
b=29−a=29−7=22b = 29 - a = 29 - 7 = 22

This corresponds to Option (c).

Hence, the correct option is **Option (c)**.

About This Chapter: Equations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Linear, Quadratic and Cubic Equations

This chapter covers Linear, Quadratic and Cubic Equations and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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