CHAPTER 7: POLYNOMIALS
Chapter objectives:
• know and use the remainder and factor theorems
• find factors of polynomials
• solve cubic equations
A polynomial is a function, 𝑓(𝑥), consisting of the sum of several terms that
contain different powers i.e.:
𝑓 (𝑥 ) = 𝑎k + 𝑎/𝑥 + 𝑎0 𝑥 0 + 𝑎] 𝑥 ] + ⋯ + 𝑎• 𝑥 •
Examples of polynomials are:
𝑓 (𝑥 ) = 𝑥 + 3
𝑓 (𝑥 ) = 2𝑥 0 − 3𝑥 + 1
𝑓 (𝑥 ) = 5𝑥 ] + 3𝑥
𝑓 (𝑥 ) = 2 + 3𝑥 n + 6𝑥 ª − 𝑥 /k
The order of a polynomial is the highest power in the polynomial so the order
of the polynomial 1 + 𝑥 ] − 𝑥 ¯ would be 9. Likewise, polynomials are named
according the highest power, so polynomials of the order 1 would be linear,
of the order 2 quadratic, of the order 3 cubic, etc.
The factor and remainder theorem
The remainder theorem states that when a polynomial 𝑓(𝑥) is divided by a
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linear function (𝑎𝑥 − 𝑏) the remainder is equal to 𝑓 x y.
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, A factor exactly divides a number, i.e. it leaves a remainder of 0. From this,
the factor theorem is derived which states that if (𝑎𝑥 − 𝑏) is a factor of the
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polynomial 𝑓(𝑥), then 𝑓 x y = 0.
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Example 7.1
It is given that when 𝑔(𝑥 ) = 𝑥 ] + 𝑎𝑥 0 + 2𝑥 − 36 is divided by 𝑥 + 2 the
remainder is 4. Find the value of 𝑎.
SOLUTION
The remainder when 𝑔(𝑥 ) is divided by 𝑥 + 2 is 𝑔(−2).
Also this remainder = 4
→ 𝑔(−2) = 4
→ (−2)] + 𝑎(−2)0 + 2(−2) − 36 = 4
→ −48 + 4𝑎 = 4
→ 4𝑎 = 52
∴ 𝑎 = 13
Example 7.2
The polynomial 𝑓 (𝑥 ) = 𝑎𝑥 ] − 15𝑥 0 + 𝑏𝑥 − 2 has a factor 2𝑥 − 1 and a
remainder of 5 when divided by 𝑥 − 1. Find the values of 𝑎 and 𝑏.
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Chapter objectives:
• know and use the remainder and factor theorems
• find factors of polynomials
• solve cubic equations
A polynomial is a function, 𝑓(𝑥), consisting of the sum of several terms that
contain different powers i.e.:
𝑓 (𝑥 ) = 𝑎k + 𝑎/𝑥 + 𝑎0 𝑥 0 + 𝑎] 𝑥 ] + ⋯ + 𝑎• 𝑥 •
Examples of polynomials are:
𝑓 (𝑥 ) = 𝑥 + 3
𝑓 (𝑥 ) = 2𝑥 0 − 3𝑥 + 1
𝑓 (𝑥 ) = 5𝑥 ] + 3𝑥
𝑓 (𝑥 ) = 2 + 3𝑥 n + 6𝑥 ª − 𝑥 /k
The order of a polynomial is the highest power in the polynomial so the order
of the polynomial 1 + 𝑥 ] − 𝑥 ¯ would be 9. Likewise, polynomials are named
according the highest power, so polynomials of the order 1 would be linear,
of the order 2 quadratic, of the order 3 cubic, etc.
The factor and remainder theorem
The remainder theorem states that when a polynomial 𝑓(𝑥) is divided by a
7
linear function (𝑎𝑥 − 𝑏) the remainder is equal to 𝑓 x y.
6
133
, A factor exactly divides a number, i.e. it leaves a remainder of 0. From this,
the factor theorem is derived which states that if (𝑎𝑥 − 𝑏) is a factor of the
7
polynomial 𝑓(𝑥), then 𝑓 x y = 0.
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Example 7.1
It is given that when 𝑔(𝑥 ) = 𝑥 ] + 𝑎𝑥 0 + 2𝑥 − 36 is divided by 𝑥 + 2 the
remainder is 4. Find the value of 𝑎.
SOLUTION
The remainder when 𝑔(𝑥 ) is divided by 𝑥 + 2 is 𝑔(−2).
Also this remainder = 4
→ 𝑔(−2) = 4
→ (−2)] + 𝑎(−2)0 + 2(−2) − 36 = 4
→ −48 + 4𝑎 = 4
→ 4𝑎 = 52
∴ 𝑎 = 13
Example 7.2
The polynomial 𝑓 (𝑥 ) = 𝑎𝑥 ] − 15𝑥 0 + 𝑏𝑥 − 2 has a factor 2𝑥 − 1 and a
remainder of 5 when divided by 𝑥 − 1. Find the values of 𝑎 and 𝑏.
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