The Factor And Remainder Theorems
Gbogbo ọrọ náà
How can you tell whether a linear expression divides exactly into a polynomial without carrying out the full long division? The factor theorem and the remainder theorem give you a direct shortcut: substitute a single value, and the answer tells you everything you need. These two results are the key tools for breaking cubic and higher-degree polynomials into simpler factors.
In this topic you will learn to test whether a given linear expression is a factor, find the remainder from any linear divisor, and combine these techniques with algebraic division to factorise cubics completely. These skills appear frequently on both papers of the Edexcel 4PM1 exam and connect directly to solving polynomial equations.
Ebumnobi
- Know that if f(x) = 0 when x = a, then (x - a) is a factor of f(x)
- Factorise cubic expressions when a factor has been provided
- Determine the remainder when the polynomial f(x) is divided by (ax + b) or (ax - b)
Akọmọ Ojú-ẹkọ
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Budata ngwa Green Bridge CBT na ekwentị maọbụ kọmputa gị iji nweta akwụkwọ ndụmọdụ zuru oke, ajụjụ mmụta, na ndị ọzọ.
Ayẹwo Ẹkọ
Ekele diri gi maka imecha ihe karịrị na The Factor And Remainder Theorems. Ugbu a na ị na-enyochakwa isi echiche na echiche ndị dị mkpa, ọ bụ oge iji nwalee ihe ị ma. Ngwa a na-enye ụdị ajụjụ ọmụmụ dị iche iche emebere iji kwado nghọta gị wee nyere gị aka ịmata otú ị ghọtara ihe ndị a kụziri.
Ị ga-ahụ ngwakọta nke ụdị ajụjụ dị iche iche, gụnyere ajụjụ chọrọ ịhọrọ otu n’ime ọtụtụ azịza, ajụjụ chọrọ mkpirisi azịza, na ajụjụ ede ede. A na-arụpụta ajụjụ ọ bụla nke ọma iji nwalee akụkụ dị iche iche nke ihe ọmụma gị na nkà nke ịtụgharị uche.
Jiri akụkụ a nke nyocha ka ohere iji kụziere ihe ị matara banyere isiokwu ahụ ma chọpụta ebe ọ bụla ị nwere ike ịchọ ọmụmụ ihe ọzọ. Ekwela ka nsogbu ọ bụla ị na-eche ihu mee ka ị daa mba; kama, lee ha anya dị ka ohere maka ịzụlite onwe gị na imeziwanye.
- If f(x) = x^3 - 3x^2 + 2x - 6 and f(3) = 0, which of the following is a factor of f(x)? A) (x + 3) B) (x - 3) C) (x - 6) D) (x + 1) Answer: B
- When f(x) = x^3 + x^2 - 2x + 1 is divided by (x - 1), the remainder is: A) 0 B) 1 C) -1 D) 3 Answer: B
- Which value of x should you substitute to test whether (x + 4) is a factor of f(x)? A) x = 4 B) x = -4 C) x = 1/4 D) x = -1/4 Answer: B
- The polynomial f(x) = 2x^3 - x^2 + kx + 6 has (x - 2) as a factor. What is k? A) -5 B) -7 C) 5 D) 7 Answer: A
- If the remainder when f(x) is divided by (x - a) is equal to zero, then: A) a is a coefficient of f(x) B) (x - a) is a factor of f(x) C) f(x) has degree a D) f(x) has no real roots Answer: B
Ọ dị na ngwa Green Bridge
Budata ngwa Green Bridge CBT na ekwentị maọbụ kọmputa gị iji nweta akwụkwọ ndụmọdụ zuru oke, ajụjụ mmụta, na ndị ọzọ.
Ọ dị na ngwa Green Bridge
Budata ngwa Green Bridge CBT na ekwentị maọbụ kọmputa gị iji nweta akwụkwọ ndụmọdụ zuru oke, ajụjụ mmụta, na ndị ọzọ.
Àwọn Ìwé Tó Dáa Láti Kà
Meecha Ajụjụ Ule Ọmarịcha
Ị chọrọ ime ajụjụ ule ọmarịcha gbasara The Factor And Remainder Theorems? Budata ngwa Green Bridge CBT iji nweta ajụjụ ule ọmarịcha na nyocha zuru ezu gbasara isiokwu a.
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