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Non-polynomial long division example:
When a polynomial function P(x) is divided by (x-b)→ P(b) = R
When a polynomial P(x) is divided by (ax-b) → P(b/a) = R
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a² + 2ab + b² = (a+b)²
a² - 2ab + b² = (a-b)²
a² - b² = (a+b)(a-b)
a³ - b³ = (a-b)(a²+ab+b²)
a³ + b³ = (a+b)(a² - ab + b²)
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Factor theorem states that…
x-b is a factor of a polynomial p(x) if and only if → p(b)=0
ax-b is a factor of a polynomial p(x) if and only if → p(b/a) = 0
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If x-b is a factor of polynomial p(x) with leading coefficient 1 and remaining coefficients that are integers then…
b is a factor of the constant
e.g.
If polynomial p(x) has integer coefficients and x = b/a is a rational zero of p(x), then….
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e.g.
The real roots of a polynomial equation can be found by setting the equation to zero, y=0
If the polynomial is factorable, the roots can be determined by factoring first fully and then setting each individual factor bracket to zero
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f(x) = a(x-a₁)(x-a₂)(x-a₃)…(x-aₙ)
a is different for each member of the family
The equation for a family can be determined from the zeros (x int) and a specific member of the family can be determined if given a specific point on the function
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Isolate for x
Treat the inequality almost like an equal sign
e.g.
The difference between = and an inequality: If we multiply or divide by a negative number in an inequality, we must flip or reverse the inequality
e.g.
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Factor
Solve the brackets
Draw a number line to identify the intervals
Create a chart
Pick a random number in between the intervals (not including the written numbers as those aren’t included, we used circle brackets) and substitute it into the variable in each factor row put in → write the sign of the number that you get
Determine the final sign by multiplying all of the signs down to get the overall functions sign
Write your final X E statement based on what your looking for, and remember to put U in between intervals if there is more than one
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