Page 25 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

Example 14

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1 If P(x) = 2x + 4x – x + 3 and Q(x) = 3x + x – 1, find
(a) P(x) + Q(x),
(b) P(x) – Q(x),
(c) P(x) · Q(x).
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Solution: (a) P(x) + Q(x) = 2x + 4x – x + 3 + 3x + x – 1
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= 2x + 7x + 2
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(b) P(x) – Q(x) = 2x + 4x – x + 3 – (3x + x – 1)
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= 2x + 4x – x + 3 – 3x – x + 1
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= 2x + x – 2x + 4
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(c) P(x) · Q(x) = (2x + 4x – x + 3) · (3x + x – 1)
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= 2x (3x + x – 1) + 4x (3x + x – 1)
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– x(3x + x – 1) + 3(3x + x – 1)
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= 6x + 2x – 2x + 12x + 4x – 4x
– 3x – x + x + 9x + 3x – 3
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= 6x + 14x – x + 4x + 4x – 3 Polynomial of degree 5.
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Note: If P(x) is a polynomial of degree m and Q(x) is a polynomial of degree n, then P(x) · Q(x) is a polynomial
of degree (m + n).
For the division of two polynomials, the long division method may be used, as shown in the following
example.
Example 15
Determine the quotient and remainder when 2x – 7x – 9x + 38 is divided by (x – 3).
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Solution: 2x – x – 12
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x – 3 2x – 7x – 9x + 38
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2x – 6x
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– x – 9x
– x + 3x
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– 12x + 38
– 12x + 36
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Using the long division method, the quotient is 2x – x – 12 and remainder 2.
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From Example 15 above, we know that a polynomial P(x), of degree m, may be divided by another polynomial,
Q(x), of degree n, only if m  n, m, n ∈ Z . If P(x) is divisible by Q(x) exactly, i.e. without any remainder,
+
then the quotient is another polynomial of degree (m – n).
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01a STPM Math T T1.indd 22 3/28/18 4:20 PM