SME
Grade 11 Math · Problem type
Divide by a quadratic
Use this for “Divide x³ + 2x² − 5x + 7 by x² − x + 2.”
Chart
Zero: x ≈ −3.8003
This is an example. Type your own values to start a new problem.
(
a
x³
+
b
x²
+
(
c
)
x
+
d
)
÷
(x²
+
(
p
)
x
+
q
)
A
Quotient’s x coefficient
worked out (not typed)
B
Quotient’s constant
worked out (not typed)
C
Remainder’s x coefficient
worked out (not typed)
D
Remainder’s constant
worked out (not typed)
Formulas
A
=
a
1 = 1
B
=
b
−
p
×
A
3 = 2 − (−1) × 1
C
=
c
−
q
×
A
−
p
×
B
−4 = −5 − 2 × 1 − (−1) × 3
D
=
d
−
q
×
B
1 = 7 − 2 × 3
Assumptions
•
Divide the leading terms, multiply the divisor by the result, subtract, and repeat, as in long division of numbers.
•
Stop when what is left has a lower degree than the divisor, x² + px + q: that is the remainder.
•
Check: divisor × quotient + remainder gives back P(x).
Step-by-step
WE KNOW
a = 1, b = 2, c = −5, d = 7, p = −1, q = 2
FIND
quotient’s x coefficient (A), quotient’s constant (B), remainder’s x coefficient (C), remainder’s constant (D)
Step 1 · Find quotient’s x coefficient (A)
Use A = a
Divide
the
leading
terms:
ax³
x
2
=
ax.
x
3
x
2
=
x
A
=
1
Step 2 · Find quotient’s constant (B)
Use B = b − p × A
Take
Ax(
x
2
+
px
+
q
)
away:
the
x
2
term
left,
divided
by
x
2
,
is
B
.
B
=
2
−
(−1)
×
1
x
3
+
2x²
−
5x
+
7
−
x(
x
2
−
x
+
2)
=
3x²
−
7x
+
7
B
=
3
Step 3 · Find remainder’s x coefficient (C)
Use C = c − q × A − p × B
Take
B
(
x
2
+
px
+
q
)
away
too:
the
x
term
left
belongs
to
the
remainder.
C
=
−5
−
2
×
1
−
(−1)
×
3
C
=
−5
−
2
−
(−3)
C
=
−4
Step 4 · Find remainder’s constant (D)
Use D = d − q × B
The constant left after both subtractions.
D
=
7
−
2
×
3
x
3
+
2x²
−
5x
+
7
=
(
x
2
−
x
+
2)(x
+
3)
+
(−4x
+
1)
D
=
1
Check
Put the numbers back in. Both sides should match.
1 = 1 ✓
3 = 2 − (−1) × 1 ✓
−4 = −5 − 2 × 1 − (−1) × 3 ✓
1 = 7 − 2 × 3 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Adding, subtracting and multiplying polynomials
Factoring polynomials: GCF, trinomials and special products