Flashcards · 12 cards

Polynomials & algebraic division

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Rule
In $2x^3 - 5x^2 + x - 7$ : the degree, the leading term, the constant?
Answer
Degree 3, leading term $2x^3$ , constant $-7$ . The coefficient of $x^2$ is $-5$ .
2x³ − 5x² + x − 7
leading termcoefficient of x²constant
Highest power 3: degree 3 (a cubic)
Parts of a polynomialLeading term 2x³, coefficient of x² is −5, constant −7, degree 3
From the lesson: Multiplying and dividing polynomials
Rule
How do you multiply $(x - 2)(x^2 + 3x - 1)$ ?
Answer
Every term times every term, then collect like powers: $x^3 + 3x^2 - x - 2x^2 - 6x + 2 = x^3 + x^2 - 7x + 2$ .
× x² 3x −1
x x³ 3x² −x
−2 −2x² −6x +2
Every term times every termEach cell is a row term times a column term; add the cells, collecting like powers
From the lesson: Multiplying and dividing polynomials
Rule
How are the dividend, divisor, quotient and remainder related?
Answer
Dividend = divisor × quotient + remainder, just as $17 = 5 \times 3 + 2$ .
17 = 5 × 3 + 2
f(x) = divisor × quotient + remainder
The remainder has a lower degree than the divisor
Dividend, divisor, quotient, remainderThe same relationship as for numbers
From the lesson: Multiplying and dividing polynomials
Know it
In long division, what do you do to the row you subtract?
Answer
Change the sign of every term in it: $(2x^2 + 5x) - (2x^2 + 6x) = -x$ .
From the lesson: Multiplying and dividing polynomials
Rule
The remainder theorem?
Answer
Dividing $f(x)$ by $x - a$ leaves the remainder $f(a)$ . For $x + 3$ , work out $f(-3)$ .
Remainder = f(a)The height of y = f(x) at x = a is the remainder on dividing by x − a
From the lesson: The remainder and factor theorems
Rule
The factor theorem?
Answer
$x - a$ is a factor of $f(x)$ exactly when $f(a) = 0$ .
Factor: f(a) = 0The curve meets the x-axis at x = a, so x − a is a factor
From the lesson: The remainder and factor theorems
Know it
The factor $x + 2$ belongs to which root?
Answer
$x = -2$ : the sign changes. Put it in brackets when you substitute: $(-2)^3 = -8$ , $(-2)^2 = 4$ .
From the lesson: The remainder and factor theorems
Rule
How do you build the polynomial with zeros $-2$ , $-1$ and $3$ ?
Answer
Turn each zero into a factor, $(x + 2)(x + 1)(x - 3)$ , then multiply out.
Three rootsy = (x − p)(x − q)(x − r) crosses the x-axis at p, q and r
From the lesson: Solving cubic equations
Rule
What does a squared factor $(x - a)^2$ do to the graph?
Answer
The curve touches the $x$ -axis at $a$ instead of crossing it: a repeated root.
A repeated rootA squared factor (x − a)² makes the curve touch the axis at a
From the lesson: Solving cubic equations
Know it
How do you solve a cubic equation?
Answer
Find one root by trying small factors of the constant term (the factor theorem). Divide by that factor to get a quadratic, then solve the quadratic.
From the lesson: Solving cubic equations
Rule
Factorise $a^3 + b^3$ and $a^3 - b^3$ .
Answer
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ .
a³ − b³ = (a − b)(a² + ab + b²)
a³ + b³ = (a + b)(a² − ab + b²)
same sign as the question, then the opposite sign, then always +
Sum and difference of cubesA linear bracket times a quadratic bracket, with the signs as marked
From the lesson: Solving cubic equations
Which method?
JAMB 2012 · UTME · Q12
Find the remainder when $2x^3 - 11x^2 + 8x - 1$ is divided by $x + 3$ .
How do you find the remainder without dividing?
Answer
The remainder theorem: dividing by $x + 3$ leaves $f(-3)$ . Put $-3$ in brackets: $(-3)^3 = -27$ and $(-3)^2 = 9$ .
From the lesson: The remainder and factor theorems Try the question

×	x²	3x	−1
x	x³	3x²	−x
−2	−2x²	−6x	+2