Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
A polynomial is a combination of positive integer powers xn:
This can also be expressed as a summation:
The degree of a polynomial is the highest power of x that appears, which is n for the above polynomials.
The degree of a product of polynomials is the sum of the degrees:
Each term ak​xk in a polynomial is associated with a specific power of x. We say that
and
When we have a product of polynomials, it is possible to find the coefficient of a specific term without expanding the entire product. To do this, we consider which terms from the first polynomial multiplies with which term from the second polynomial to give the desired power of x.
If two polynomials are equal for all x∈R (not just an intersection), then all their coefficients are equal:
If one polynomial P(x) is the product of two polynomials D(x) and Q(x), then we can say that D(x) is a divisor of P(x).
In general polynomials are not perfect divisors of each other, but instead leave a remainder:
Here, P(x) is the dividend, D(x) is the divisor, Q(x) is the quotient and R(x) is the remainder.
The quotient is the "largest" possible polynomial that can multiply D(x) without exceeding the degree of P(x).
The quotient and remainder of a polynomial division can be found using long division, an iterative process where we cancel out the highest powers of the dividend one by one.
The steps in the above division are
Compare the leading terms 11x2 and x, and write 11x in the space for the quotient.
Subtract 11x⋅(x−5)
Compare the new leading term 29x to x, and write +29 in the space for the divisor.
Subtract 29⋅(x−5), leaving a constant of 110.
Stop the division as the leading power is not smaller than x1.
When the divisor is linear, the remainder is constant.
Polynomial long division stops when the remainder cannot be further divided by the divisor, that is the degree of the remainder is less than that of the divisor.
And since the degree of a product of polynomials is the sum of their degrees, the degree of the quotient and divisor must add up to the degree of the original polynomial. So if
then
The degree of R is less than the degree of D. 🚫
The degree of Q plus the degree of D equals the degree of P 🚫.
The polynomial factor theorem states that
The remainder theorem states that
The coefficient a of the leading term (the one with the highest ) determines the broad shape of its graph:
Cubic
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Quartic
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The curve of a polynomial touches the x-axis at each real root. The shape of the curve at these points depends on whether the root is unique or repeated:
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A polynomial with degree n has n roots. These roots are not necessarily distinct, and not always real.
For a polynomial's roots:
For a polynomial with real coefficients, if a complex number z is a root, then the conjugate z⋆ must also be a root.