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Nata [24]
3 years ago
9

When a polynomial f(x) is divided by 2x-3,the quotient is x^2-x+2 and the remainder is -1. Find the f(x)​

Mathematics
1 answer:
joja [24]3 years ago
3 0

Answer:

f(x) = 2x³ - 5x² + 7x - 7

Step-by-step explanation:

In the division statement: m ÷ n = q + \frac{r}{n}

  • m is the dividend
  • n is the divisor
  • q is the quotient
  • r is the remainder
  • m = q × n + r

Let us use the fact above to solve the question

∵ f(x) is divided by (2x - 3), the quotient is x² - x + 2 and the remainder is -1

∴ f(x) is the dividend ⇒ m

∴ (2x - 3) is the divisor ⇒ n

∴ (x² - x + 2) is the quotient ⇒ q

∴ -1 is the remainder ⇒ r

→ Use the rule above to find f(x)

∵ f(x) = (x² - x + 2) × (2x - 3) + -1

∴ f(x) = (x² - x + 2)(2x - 3) - 1

→ Multiply the 2 brackets at first

∵ (x² - x + 2)(2x - 3) = x²(2x) + x²(-3) + -x(2x) + -x(-3) + 2(2x) + 2(-3)

∴ (x² - x + 2)(2x - 3) = 2x³ - 3x² - 2x² + 3x + 4x - 6

→ Add the like terms

∴ (x² - x + 2)(2x - 3) = 2x³ + (-3x² - 2x²) + (3x + 4x) - 6

∴ (x² - x + 2)(2x - 3) = 2x³ + (-5x²) + 7x - 6

∴ (x² - x + 2)(2x - 3) = 2x³ - 5x² + 7x - 6

→ Substitute it in f(x)

∴ f(x) = 2x³ - 5x² + 7x - 6 - 1

→ Add the like term

∵ f(x) = 2x³ - 5x² + 7x + (- 6 - 1)

∴ f(x) = 2x³ - 5x² + 7x + (-7)

∴ f(x) = 2x³ - 5x² + 7x - 7

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