Flashcards · 13 cards

Polynomials & quadratic roots

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Rule
How does synthetic division by $x - 2$ work?
Answer
Write 2 on the left and the coefficients along the top (0 for any missing power). Bring the first down; then multiply by 2 and add, column by column. The last number is the remainder.
2 1 2 −5 1
2 8 6
1 4 3 7
quotient x² + 4x + 3, remainder 7
Bring down the 1. Then × 2, write it under the next number, and add
Synthetic division(x³ + 2x² − 5x + 1) ÷ (x − 2): quotient x² + 4x + 3, remainder 7
From the lesson: Dividing, factorising and finding unknowns
Rule
Dividing $f(x)$ gives a quotient and a remainder. How are they related, and how does that find unknowns?
Answer
$f(x) = \text{divisor} \times \text{quotient} + \text{remainder}$ . Multiply out the right side and compare coefficients.
17 = 5 × 3 + 2
f(x) = divisor × quotient + remainder
The remainder has a lower degree than the divisor
Division as multiplicationf(x) = divisor × quotient + remainder
From the lesson: Dividing, factorising and finding unknowns
Know it
A cubic has zeros $-1$ , $\frac12$ and $-\frac53$ . What are its factors?
Answer
$(x + 1)$ , $(2x - 1)$ and $(3x + 5)$ . Clear the fraction to get each bracket; the zeros are not the factors.
From the lesson: Dividing, factorising and finding unknowns
Rule
The sum and product of the roots of $ax^2 + bx + c = 0$ ?
Answer
$\alpha + \beta = -\dfrac ba$ and $\alpha\beta = \dfrac ca$ .
ax² + bx + c = 0, roots α and β
α + β = −b⁄a αβ = c⁄a
The equation is x² − (sum)x + (product) = 0
Sum and product of the rootsSum −b/a, product c/a
From the lesson: The roots α and β
Rule
$\alpha^2 + \beta^2$ in terms of the sum and product?
Answer
$(\alpha + \beta)^2 - 2\alpha\beta$ .
(α + β)² = α² + 2αβ + β²Take away the two rectangles: α² + β² = (α + β)² − 2αβ
From the lesson: The roots α and β
Know it
$(\alpha - \beta)^2$ in terms of the sum and product?
Answer
$(\alpha + \beta)^2 - 4\alpha\beta$ . Four, not two: two is for $\alpha^2 + \beta^2$ .
From the lesson: The roots α and β
Know it
$\alpha^3 + \beta^3$ in terms of the sum and product?
Answer
$(\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$ .
From the lesson: The roots α and β
Know it
$\dfrac1\alpha + \dfrac1\beta = \,?$
Answer
$\dfrac{\alpha + \beta}{\alpha\beta}$ .
From the lesson: The roots α and β
Know it
The quadratic equation whose roots are $p$ and $q$ ?
Answer
$x^2 - (p + q)x + pq = 0$ : $x^2$ minus the sum times $x$ , plus the product.
From the lesson: The roots α and β
Rule
The condition for real roots?
Answer
$b^2 - 4ac \ge 0$ . Equal roots: $= 0$ . No real roots: $< 0$ .
What b² − 4ac tells youReal roots means two different or equal: b² − 4ac ≥ 0
From the lesson: Completing the square, the discriminant and inequalities
Which method?
WAEC 2022 · Paper 2 · Q2
If $\alpha$ and $\beta$ are the roots of $2x^2 - x - 2 = 0$ , find the value of $\alpha^3 + \beta^3$ .
Value of $\alpha^3 + \beta^3$
How do you avoid solving for $\alpha$ and $\beta$ ?
Answer
Find $\alpha + \beta = \frac12$ and $\alpha\beta = -1$ from the coefficients, then use $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$ .
From the lesson: The roots α and β Try the question
Which method?
WAEC 2011 · Paper 2 · Q1
If $f(x) = 6x^3 + 13x^2 + 2x - 5$ and $f(-1) = 0$ , find the factors of $f(x)$ .
You are told $f(-1) = 0$ . What next?
Answer
By the factor theorem, $(x + 1)$ is a factor. Divide by it (synthetic division), then factorise the quadratic that is left.
From the lesson: Dividing, factorising and finding unknowns Try the question
Which method?
WAEC 2019 · Paper 2 · Q3 (b)
Find the range of values of $n$ for which $3 + 14n - 5n^2 \le 0$ .
After finding the roots $-\frac15$ and $3$ , which side do you want?
Answer
The $n^2$ term is negative, so the curve opens down. It is $\le 0$ outside the roots: $n \le -\frac15$ or $n \ge 3$ .
From the lesson: Completing the square, the discriminant and inequalities Try the question

2	1	2	−5	1
		2	8	6
	1	4	3	7