SME
Grade 11 Math · Algebra 2
Polynomial equations: sums and differences of cubes, rational roots and the fundamental theorem of algebra
Use this for “Solve 2x³ − 3x² − 11x + 6 = 0” with the rational root theorem and synthetic division.
Chart
Zeros: r = −2, r = 0.5, r = 3
f(3) = 0
Quotient’s x coefficient: 3
Quotient’s constant: −2
Remainder: 0
This is an example. Type your own values to start a new problem.
a
x³
+
(
b
)
x²
+
(
c
)
x
+
d
=
0
r
Root tested
example
q₁
Quotient’s x coefficient
worked out (not typed)
q₀
Quotient’s constant
worked out (not typed)
R
Remainder
worked out (not typed)
x₂
Larger other root
worked out (not typed)
x₃
Smaller other root
worked out (not typed)
Formulas
q₁
=
b
+
r
×
a
3 = −3 + 3 × 2
q₀
=
c
+
r
×
q₁
−2 = −11 + 3 × 3
R
=
d
+
r
×
q₀
0 = 6 + 3 × (−2)
x₂
=
−
q₁
+
√
q₁
2
− 4 ×
a
×
q₀
2 ×
a
,
when
R
=
0
1
2
=
(−3
+
√(
3
2
−
4
×
2
×
(−2)))
÷
(2
×
2),
when
0
=
0
x₃
=
−
q₁
−
√
q₁
2
− 4 ×
a
×
q₀
2 ×
a
,
when
R
=
0
−2
=
(−3
−
√(
3
2
−
4
×
2
×
(−2)))
÷
(2
×
2),
when
0
=
0
Assumptions
•
Any rational root is ±(a factor of d) ÷ (a factor of a): test these candidates as r.
•
Synthetic division by x − r leaves remainder 0 exactly when r is a root.
•
The quotient is the quadratic ax² + q₁x + q₀, which the quadratic formula finishes.
Step-by-step
WE KNOW
a = 2, b = −3, c = −11, d = 6, r = 3
FIND
quotient’s x coefficient (q₁), quotient’s constant (q₀), remainder (R), larger other root (x₂), smaller other root (x₃)
Step 1 · Find quotient’s x coefficient (q₁)
Use q₁ = b + r × a
Candidates:
±
1, 2, 3, 6
1, 2
=
±1,
±2,
±3,
±6,
±
1
2
,
±
3
2
;
try
r
=
3.
Bring
down
a
,
multiply
by
r
and
add
to
b
.
q₁
=
−3
+
3
×
2
q₁
=
−3
+
6
q₁
=
3
Step 2 · Find quotient’s constant (q₀)
Use q₀ = c + r × q₁
Multiply
by
r
and
add
to
c
.
q₀
=
−11
+
3
×
3
q₀
=
−11
+
9
q₀
=
−2
Step 3 · Find remainder (R)
Use R = d + r × q₀
The
last
sum
is
the
remainder:
0
means
x
−
r
is
a
factor.
R
=
6
+
3
×
(−2)
3
2
−3
−11
6
6
9
−6
2
3
−2
0
R
=
0
Step 4 · Find larger other root (x₂)
Use x₂ = (−q₁ + √(q₁² − 4 × a × q₀)) ÷ (2 × a), when R = 0
The quadratic formula on the quotient, with the plus sign.
x₂
=
−
q₁
+
√
q₁
2
− 4 ×
a
×
q₀
2 ×
a
x₂
=
(−3
+
√(
3
2
−
4
×
2
×
(−2)))
÷
(2
×
2)
x₂
=
−3 +
√
25
4
x₂
=
−3 + 5
4
x₂
=
1
2
Step 5 · Find smaller other root (x₃)
Use x₃ = (−q₁ − √(q₁² − 4 × a × q₀)) ÷ (2 × a), when R = 0
The same formula with the minus sign.
x₃
=
−
q₁
−
√
q₁
2
− 4 ×
a
×
q₀
2 ×
a
x₃
=
(−3
−
√(
3
2
−
4
×
2
×
(−2)))
÷
(2
×
2)
x₃
=
−3 −
√
25
4
x₃
=
−3 − 5
4
(x
−
3)(2x²
+
3x
−
2)
=
0;
x
=
3,
1
2
,
−2
x₃
=
−2
Check
Put the numbers back in. Both sides should match.
3 = −3 + 3 × 2 ✓
−2 = −11 + 3 × 3 ✓
0 = 6 + 3 × (−2) ✓
1
2
=
−3 +
√
25
4
✓
−2
=
−3 −
√
25
4
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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