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Multiplying by (x – 1)

When multiplying polynomials in Algebra, we often encounter problems with (x – 1) as a multiplier. Our previous post on multiplying numbers by 9 will help us easily multiply any polynomial by (x – 1).

Recall that if we want to multiply 1458 by 9, we will use the following steps:

  1. Copy the first digit of the multiplicand as the 1st digit of the answer – 1
  2. Subtract the first from the next digit in the multiplicand – (4 – 1 = 3)
  3. Repeat step 2 starting from the second digit up to the penultimate digit of the multiplicand (5 – 4 = 1) and (8 – 5 = 3)
  4. Write down the last digit as a negative number which is the same as subtracting it from 0.
  5. Express the answer in the conventional format.

1 4 5 8 x 9 = 1313(8) = 13,122

If we want to multiply (2x^3 – 5x^2 + 3x + 4)(x – 1) we can use similar steps.

  1. Copy the coefficient of the first term of the multiplicand as the coefficient of the first term of the answer and increase the exponent by one: 2x^4.
  2. Subtract the coefficient of the first term from the coefficient of the second term of the multiplicand to get the coefficient of the second term of the answer: (- 5 – 2) = – 7.
  3. Repeat step 2 to get the coefficients of the succeeding terms of the answer until the penultimate term of the multiplicand: [3 – (- 5)] = 8 and (4 – 3) = 1
  4. Write down the negative of the last term of the multiplicand as the last term of the answer: 4 → -4  

The sum of the coefficients of the answer must be zero.

(2x^3 – 5x^2 + 3x + 4)(x – 1) = 2x^4 -7x^3 + 8x^2 + x – 4; check: 2 – 7 + 8 + 1 – 4 = 0

More examples:

(3 – 2x + 3x^2 – 5x^3 + 4x^4)(x – 1) → the multiplicand should be in standard form.

 (3 – 2x + 3x^2 – 5x^3 + 4x^4)(x – 1)  = (4x^4 – 5x^3 + 3x^2 – 2x + 3)(x – 1)

                                                         = 4x^5 + (-5 – 4)x^4 + [3 –(-5)]x^3 + (- 2 – 3)x^2 + [3 –(-2)x – 3   

                                                      = 4x^5 -9x^4 + 8x^3 – 5x^2 + 5x – 3         

(2x^4 + 3x^2 – 5x + 2)(x – 1) multiplicand lacking in x^3 term; insert a place holder

(2x^4 + 0x^3 + 3x^2 – 5x + 2)(x – 1) = 2x^5 + (0 – 2)x^4 + (3 – 0)x^3 + (-5 – 3)x^2 + [2 –(-5)x – 2

                                                            = 2x^5 – 2x^4 + 3x^3 – 8x^2 + 7x – 2

Try multiplying the following by (x – 1):

  1. (x – 4)
  2. (2x + 3)
  3. (3x – 2)
  4. (x^2 + 3x + 2)
  5. (2x^2 – x + 3)
  6. (4x^2 + 3x – 2)
  7. (x^3 – 4x^2 + 3x – 2)
  8. (5x^3 + 2x^2 – x + 3)
  9. (2x^4 – 5x^3 – 2x^2 + x – 3)  
  10. (3x^4 – x^3 + 4x^2 + 3x – 2)

Answers to previous post’s exercises:

  1. 35 x 8 = 3(1)(10) = 3(20) = 280
  2. 35 x 7 = 3(4)(15) = 3(55) = 245
  3. 53 x 8 = 5(7)(6) = 424
  4. 53 x 7 = 5(12)(9)= 4(29) = 371  
  5. 251 x 8 = 21(9)(2) = 2008  
  6. 251 x 7 = 2(1)(16)(3) = 2(243) = 1757
  7. 4253 x 8 = 4(6)1(7)(6) = 34,024
  8. 4253 x 7 = 4(10)(1)(12)(9) =30(229) = 29,711
  9. 7643 x 8 = 7(8)(8)(5)(6) = 7(8856) = 61,144
  10. 7643 x 7 = 7(15)(14)(9)(9)= 6(6499) = 53,501
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