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Multiplying by (x – a)

In our MATH-Inic Newsletter issue during the first week of April, we discussed multiplication by (x – 1), which is based on multiplying by 9. In our previous issues, we also discussed multiplication by 8 and 7. We are now ready to discuss the general technique of multiplying by (x – a) where a is any positive number.

Let us recall the rule for multiplication by 8 and see how it is applied in 1342 x 8:

  1. Copy the first digit of the multiplicand as the first digit of the product: 1342 x 8 = 1…
  1. Starting from the first to the penultimate digit, subtract twice each digit from the next one to the right. Write down any negative difference in bar form.

2.1) 1342 x 8 = 1(3 – 2 x 1) = 11…

2.2) 1342 x 8 = 11(4 – 2 x 3) = 11(2)…

2.3) 1342 x 8 = 112(2 – 2 x 4) = 11(2)(6) …

  1. Write the negative of twice the last digit of the multiplicand as the last digit of the answer.

3) 1342 x 8 = 11(2)(6)(2 x -2) = 11(2)(6)(4)

  1. Rewrite the answer in normal format.

11(2)(6)(4) = 10,736

Applying this method in polynomial multiplication is much easier because we can have negative coefficients and no rewriting is required.

Let us modify the steps and apply them to this example: (x^3 + 3x^2 + 4x + 2)(x – 2)

  1. Copy the first term of the multiplicand as the first term of the product but add 1 to the exponent of the variable.

(x^3 + 3x^2 + 4x + 2)(x – 2) = x^4 + …

  1. Starting from the first to the penultimate of the multiplicand, subtract twice the coefficient from the coefficient of the next term to the right. Note that multiplying by x increased the power of each term of the multiplicand by 1.   

2.1 (x^3 + 3x^3 + 4x + 2)(x – 2) = x^4 + (3 – 2(1))x^3 … = x^4 + x^3 + …

2.2 (x^3 + 3x^3 + 4x + 2)(x – 2) = x^4 + x^3 + (4 – 2(3))x^2 +… = x^4 + x^3  – 2x^2 …

2.3 (x^3 + 3x^3 + 4x + 2)(x – 2) = x^4 + x^3 – 2x^2 + (2 – 2(4))x … = x^4 + x^3 – 2x^2 – 6x…

3. The constant term of the product is twice the negative of the constant of the multiplier.

(x^3 + 3x^3 + 4x + 2)(x – 2) = x^4 + x^3 – 2x^2 – 6x + (2(– 2).

    = x^4 + x^3 – 2x^2 – 6x + – 4.

Try these following multiplications:

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

    Answers to previous exercises:

    1. 35 x 8 = 3[5 – 2 x 3][ – 2 x 5] = 3(1)(10) = 3(20) = 280   
    2. 35 x 7 = 3[5 – 3 x 3][ – 3 x 5] = 3(4)(15) = 3(55) = 245
    3. 53 x 8 = 5[3 – 2 x 5][ -2 x 3] = 5(7)(6) = 424
    4. 53 x 7 = 5[3 – 3 x 5][ -3 x 3] = 5(12)(9) = 371
    5. 251 x 8 = 2[5 – 2 x 2][1 – 2 x 5][ – 2 x 1] = 21(9)(2) = 21(92) = 2008  
    6. 251 x 7 = 2[5 – 3 x 2][1 – 3 x 5][ -3 x 1] = 2(1)(14)(3) = 2(243) = 1757
    7. 4253 x 8 = 4[2 – 2 x 4][5 – 2 x 2][3 – 2 x 5][ -2 x 3] = 4(6)1(7)(6) = 34024
    8. 4253 x 7 =  4[2 – 3 x 4][5 – 3 x 2][3 – 3 x 5][ – 3 x 3] = 4(10)(1)(12)(9) = 3(0229) = 29771
    9. 7643 x 8 = 7[6 – 2 x7][4 – 2 x 6][3 – 2 x 4][- 2 x 3] = 7(8)(8)(5)(6) = 61144
    10. 7643 x 7 = 7[6 – 3 x7][4 – 3 x 6][3 – 3 x 4][- 3 x 3] = 7(15)(14)(9)(9)= 53501
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