Solve 2015 G12 additional mathematics 4(b) in 2 steps.

Question:-Given that 3x³-13x²+18x-10=(Ax+B)(x-1)(x-2)+C for all values of x.Find the values of A, B and C .

STEP 1.

Honestly speaking,to solve such a question,the first thing you need to think of is the approach, so in our case I shall teach you how to use the expansion method which works 100% unless otherwise. Okay..! Lets expand,
I'll start with (x-1)(x-2),

• (Ax+B)[x(x-2)-1(x-2)]+C
• (Ax+B)(x²-2x-x+2)+C
• (Ax+B)(x²-3x+2)+C

Now,lets finish the expansion with (Ax+B).

• Ax(x²-3x+2)+B(x²-3x+2)+C
• Ax³-3Ax²+2Ax+Bx²-3Bx+2B+C

After expanding,group all the terms according to the powers of x,which is from x³ to x⁰ .

• Ax³-3Ax²+Bx²+2Ax-3Bx+2B+C

Factorization is now to be used,again with respect to the powers of x.

• Ax³+(B-3A)x²+(2A-3B)x+2B+C

Congratulations, you've solved the question just a few more steps and you are done.

STEP 2.

In this stage all you need to do is equate the two expressions, the original and the expanded. By original I mean the 3x3-13x2+18x-10 equation. For the two expressions to be equal the we shall equate the powers of x such that both equations will have their values of the coefficient of x equal as shown below.

• A=3 (i)
• B-3A=-13 (ii)
• 2A-3B=18 (iii)
• 2B+C=-10 (iv)

I have good news for you,equation (i) is already solved so just get the value. So now we can move on to the rest of the equations. You can use (ii) or (iii) to solve for the value of B. I'll use equation (iii).

• 2(3)-3B=18
• 6-3B=18
• 6-6-3B=18-6
• -3B=12
multiply both sides by -1/3 or divide by -3 both sides.
• B=-4

You can also try equation (ii) it should give the same answer for the value of B. Lastly,lets get the value of C by using equation (iv) with the known values of A and B though we shall only use the value of B.

• 2B+C=-10
• 2(-4)+C=-10
• -8+C=-10
• -8+8+C=-10+8
• C=-2

Therefore ,the values of A,B and C are 3,-4 and -2 respectively.

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