# 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:

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

- 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.** - 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.** - 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** - 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):

- (x – 4)
- (2x + 3)
- (3x – 2)
- (x^2 + 3x + 2)
- (2x^2 – x + 3)
- (4x^2 + 3x – 2)
- (x^3 – 4x^2 + 3x – 2)
- (5x^3 + 2x^2 – x + 3)
- (2x^4 – 5x^3 – 2x^2 + x – 3)
- (3x^4 – x^3 + 4x^2 + 3x – 2)

Answers to previous post’s exercises:

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