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Mathematics Dictionary
Dr. K. G. Shih
Factors of expressions
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Answers


Q01. Factors of a*x^2 + b*x + c by observation

  • Definition : Let a*x^2 + b*x + c = (x + m)*(x + n).
    • m*n = +c.
    • m+n = +b.
  • Example 1 : Factorize x^2 - 6*x + 8 by obsevation.
    • Note : x^2 means x to power 2. 6*x = 6 times x.
    Solution
    • Let x^2 - 6*x + 8 = (x+m)*(x+n).
      • m*n = +8.
      • m+n = -6.
    • Trials.
      • Let m = +1 and n = +8. Hence m*n = +8 and m+n = +9. Not satisfied.
      • Let m = -1 and n = +8. Hence m*n = -8 and m+n = +7. Not satisfied.
      • Let m = +1 and n = -8. Hence m*n = -8 and m+n = -7. Not satisfied.
      • Let m = -1 and n = -8. Hence m*n = +8 and m+n = -9. Not satisfied.
      • Let m = +2 and n = +4. Hence m*n = +8 and m+n = +6. Not satisfied.
      • Let m = -2 and n = +4. Hence m*n = -8 and m+n = +2. Not satisfied.
      • Let m = +2 and n = -4. Hence m*n = -8 and m+n = -2. Not satisfied.
      • Let m = -2 and n = -4. Hence m*n = +8 and m+n = -6. This is satisfied.
    • Note : We donot need do all 8 trails.
      • Since m*n = +8, we only need m*n = 4*2 or (-2)*(-4).
      • Since m+n = -8, hence we need only m = -2 and n = -4.
    • Hence we have x^2 - 6*x + 8 = (x - 2)*(x - 4).
  • Example 2 : Factorize x^2 - 4 by observation.
    • Let x^2 - 4 = (x + m)*(x + n).
      • m*n = -4.
      • m+n = +0.
    • Trials.
      • Let m = +1 and n = +4. Hence m*n = +4 and m+n = +5. Not satisfied.
      • Let m = -1 and n = +4. Hence m*n = -4 and m+n = +3. Not satisfied.
      • Let m = +1 and n = -4. Hence m*n = -4 and m+n = -3. Not satisfied.
      • Let m = -1 and n = -4. Hence m*n = +4 and m+n = -5. Not satisfied.
      • Let m = +2 and n = +2. Hence m*n = +4 and m+n = +4. Not satisfied.
      • Let m = -2 and n = +2. Hence m*n = -4 and m+n = +0. This is satisfied.
      • Let m = +2 and n = -2. Hence m*n = -4 and m+n = +0. This is satisfied.
      • Let m = -2 and n = -2. Hence m*n = +4 and m+n = -4. Not satisfied.
    • Note : One trail we get the answer. i.e. m = 2 and n = -2.
    • Hence we have x^2 - 4 = (x + 2)*(x - 2).
    • Formula : x^2 - a^2 = (x + a)*(x - a).
  • Example 3 : Factorize x^2 + 4 by observation.
    Solution
    • Let x^2 + 4 = (x + m)*(x + n).
      • m*n = +4.
      • m+n = +0.
    • Trials.
      • Let m = +1 and n = +4. Hence m*n = +4 and m+n = +5. Not satisfied.
      • Let m = -1 and n = +4. Hence m*n = -4 and m+n = +3. Not satisfied.
      • Let m = +1 and n = -4. Hence m*n = -4 and m+n = -3. Not satisfied.
      • Let m = -1 and n = -4. Hence m*n = +4 and m+n = -5. Not satisfied.
      • Let m = +2 and n = +2. Hence m*n = +4 and m+n = +4. Not satisfied.
      • Let m = -2 and n = +2. Hence m*n = -4 and m+n = +0. Not satisfied.
      • Let m = +2 and n = -2. Hence m*n = -4 and m+n = +0. Not satisfied.
      • Let m = -2 and n = -2. Hence m*n = +4 and m+n = -4. Not satisfied.
    • Hence we know that x^2 + 4 has no ral factors.
    • Note : Remeber that x^2 + a^2 has no real factors.
  • Example 4 : Factorize x^2 + 4*x + 4.
    • Let x^2 + 4*x + 4 = (x + m)*(x + n).
      • m*n = +4.
      • m+n = +4.
    • Trials.
      • Let m = +1 and n = +4. Hence m*n = +4 and m+n = +5. Not satisfied.
      • Let m = -1 and n = +4. Hence m*n = -4 and m+n = +3. Not satisfied.
      • Let m = +1 and n = -4. Hence m*n = -4 and m+n = -3. Not satisfied.
      • Let m = -1 and n = -4. Hence m*n = +4 and m+n = -5. Not satisfied.
      • Let m = +2 and n = +2. Hence m*n = +4 and m+n = +4. This is satisfied.
      • Let m = -2 and n = +2. Hence m*n = -4 and m+n = +0. Not satisfied.
      • Let m = +2 and n = -2. Hence m*n = -4 and m+n = +0. Not satisfied.
      • Let m = -2 and n = -2. Hence m*n = +4 and m+n = -4. Not satisfied.
    • Hence we know that x^2 + 4*x + 4 = (x + 2)*(x + 2) = (x + 2)^2.
    • Note : This is a formula x^2 + 2*x*y + y^2 = (x + y)^2
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Q02. Factorize a*x^2 + b*x + c by quadratic formula.
To check it has real factors or not. Use discriminant method
  • To check that it has real factor or not.
  • What is discriminant ? How to check ?
    • If a*x^2 + b*x + c, the discriminant is D = b^2 - 4*a*c.
    • If D = 0, a*x^2 + b*x + c is a perfect square.
    • If D is less than 0, a*x^2 + b*x + c has no real factors.
    • If D is greater than 0, a*x^2 + b*x + c has two real factors.
How to use quadratic formula ?
  • Quadratic formula for a*x^2 + b*x + c = (x-m)(x-n)
  • If b^2 - 4*a*c is greater than zero, we have
    • m = (-b + Sqr(b^2 - 4*a*c))/(2*a).
    • n = (-b - Sqr(b^2 - 4*a*c))/(2*a).
  • If b^2 - 4*a*c is equal to zero, we have
    • m = (-b)/(2*a).
    • n = (-b)/(2*a).
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Q03. Factors of y = F(x) by factor theory
  • Factor theory
    • If (x-a) is a factof of F(x) then F(+a) = 0.
    • If (x+a) is a factof of F(x) then F(-a) = 0.
  • Remainder theory
    • If F(x) is divided by (x-a), then F(+a) is the remainder.
    • If F(x) is divided by (x-a), then F(+a) is the remainder.
Example 1 : F(x) = x^2 + 6*x + 8, find factors by factor theory
  • Posible factors : (x+1), (x-1), (x+2), (x-2), (x+4), (x-4), (x+8), (x-8).
  • Since coefficient x is +6, we will use (x+2) as first trail.
  • If (x+2) is a factor, then F(-2) must be zero.
  • F(-2) = (-2)^2 + 6*(-2) + 8 = 4 - 12 + 8 = 0. Hence (x+2) is a factor.
  • F(-4) = (-4)^2 + 6*(-4) + 8 = 16 - 24 + 8 = 0. Hence (x+4) is a factor.
  • Hence F(x) = x^2 + 6*x + 8 = (x+2)*(x+2).
Example 2 : F(x) = x^3 + 8, find factors of F(x).
  • Since F(+1) = 1^3 + 8 = 9, hence (x-1) is not a factor.
  • Since F(-1) = (-1)^3 + 8 = 7, hence (x+1) is not a factor.
  • Since F(+2) = 2^3 + 8 = 16, hence (x-2) is not a factor.
  • Since F(-2) = (-2)^3 + 8 = 0, hence (x+2) is not a factor.
  • Since F(+4) = 4^3 + 8 = 72, hence (x-4) is not a factor.
  • Since F(-4) = (-4)^3 + 8 = -56, hence (x+4) is not a factor.
  • Since F(+8) = 8^3 + 8 = 520, hence (x-8) is not a factor.
  • Since F(-8) = (-8)^3 + 8 = -512, hence (x+8) is not a factor.
  • Only one factor (x + 2), we have to find other factors by division.
  • Hence x^3 + 8 = (x + 2)*(x^2 - x*(2) + (2)^2) = (x + 2)*(x^2-2*x+4)
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Q04. Factorize a*x^3 - d by synthetic division
  • Example : Factorize x^3 - 8.
    • We use synthtic division : Assume (x - 1) is a factor
      • +1 +0 +0 -8 | 1 (Note : for x - 1 we use 1)
      • +1 +1 +1
      • -------------
      • +1 +1 +1 -7 ......... The remaider is -7. Hence (x-1) is not a factor.
    • We use synthtic division : Assume (x - 2) is a factor
      • +1 +0 +0 -8 | 2
      • +2 +4 +8
      • -------------
      • +1 +2 +4 +0 ......... The remaider is 0. Hence (x-2) is a factor.
    • Hence x^3 - 8 = (x - 2)*(x^2 + 2*x + 4)
    • Note : (x^2 + 2*x + 4) has no real factors (Use discriminant method)
    • Formula : (x^3 - y^3) = (x + y)*(x^2 + (x^2)*(y^2) + y^2).
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Q05. Factorize a*x^3 + d by synthetic division
  • Example : Factorize x^3 + 8.
    • We use synthtic division : Assume (x + 1) is a factor
      • +1 +0 +0 -8 | -1 (Note : for x + 1 we use -1)
      • -1 +1 -1
      • -------------
      • +1 -1 +1 -9 ......... The remaider is -9. Hence (x+1) is not a factor.
    • We use synthtic division : Assume (x + 2) is a factor
      • +1 +0 +0 -8 | -2
      • -2 +4 +8
      • -------------
      • +1 -2 -4 +0 ......... The remaider is 0. Hence (x+2) is a factor.
    • Hence x^3 - 8 = (x - 2)*(x^2 + 2*x + 4)
    • Note : (x^2 + 2*x + 4) has no real factors (Use discriminant method)
    • Formula : (x^3 + y^3) = (x + y)*(x^2 - (x^2)*(y^2) + y^2).
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Q06. Factorize x^4 - 16
Solution
  • Since x^4 - 16 = x^4 - 2^4 = (x^2)^2 - (2^2)^2.
  • Hence we can use formula x^2 - y^2 = (x + y)*(x - y).
  • Hence x^4 - 16 = (x^2 + 2^2)*(x^2 - 2^2).
  • Hence x^4 - 16 = (x^2 + 2^2)*(x - 2)*(x + 2).
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Q07. Factorize x^4 + 16
Solution
  • No real factors.
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Q08. Factorize x^5 - a^5
  • x^5 - a^5 = (x - a)*(x^4 + (x^3)*a + (x^2)*(a^2) + x*(a^3) + a^4).
  • Proof
    • RSH = x*(x^4 + (x^3)*a + (x^2)*(a^2) + x*(a^3) + a^4).
    • - a*(x^4 + (x^3)*a + (x^2)*(a^2) + x*(a^3) + a^4).
    • = x^5 - a^5.
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Q09. Questions
  • 1. Factorize x^2 - 6*x + 8.
  • 2. Factorize x^2 - 7*x + 8.
  • 3. Factorize x^2 - 9*x + 8.
  • 4. Factorize x^2 - 2*x + 8.
  • 5. Factorize x^2 + 2*x + 8.
  • 6. What is remainder theory ?
  • 7. What is factor theory ?
  • 8. What is the quadratic formula ?
  • 9. What is the discriminant of a quadratic function ?
  • 10 What is quadratic function ?
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Q10. Formula
Expressions with Power 2
  • x^2 - y^2 = (x + y)*(x - y).
  • x^2 + y^2 has no real factors.
  • x^2 - 2*x*y + y^2 = (x - y)^2.
  • x^2 + 2*x*y + y^2 = (x + y)^2.
Expressions with Power 3
  • x^3 - y^3 = (x - y)*(x^2 + x*y + y^2).
  • x^3 + y^3 = (x + y)*(x^2 - x*y + y^2).
  • x^3 - 3*(x^2)*y + 3*x(y^2) - y^3 = (x - y)^3.
  • x^3 + 3*(x^2)*y + 3*x(y^2) + y^3 = (x + y)^3.
Expressions with Power 4
  • x^4 - y^4 = (x^2 + y^2)*(x - y)*(x + y).
  • x^4 + y^4 has no real factors.
  • x^4 - 4*(x^3)*y + 6*(x^2)*(y^2) - 4*(x)*(y^3) + y^4 = (x - y)^4.
  • x^4 + 4*(x^3)*y + 6*(x^2)*(y^2) + 4*(x)*(y^3) + y^4 = (x + y)^4.
Theory
  • Factor theory : If F(a) = 0 then (x-a) is factor of F(x).
  • Remainder theory : If F(x) is divided by (x-a), then F(a) is remainder.
Quadratic formula to find M and n of a*x^2 + b*x + c = (x - m)*(x - n)
  • If b^2 - 4*a*c is greater than zero, it has two real roots or factors.
    • m = (-b + Sqr(b^2 - 4*a*c))/(2*a)
    • n = (-b - Sqr(b^2 - 4*a*c))/(2*a)
  • If b^2 - 4*a*c is equal to zero, it has two real roots or factors.
    • m = (-b)/(2*a)
    • n = (-b)/(2*a) and m = n
Discriminant D = b^2 - 4*a*c
  • If D is greater than zero, it has two real factors.
  • If D is zero, it is a perfect square.
  • If D is less than zero, it has no real factors.
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