Skills Checklist | IB Math AAHL

Skill Checklist

Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.

15 Skills Available

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

Polynomial Basics

5 skills

Polynomial definition and degree

AHL 2.12

A polynomial is a combination of positive integer powers xn:

P(x)=anxn+an−1xn−1+⋯+a1x+a0

This can also be expressed as a summation:

P(x)=k=0∑nakxk

All the afactors are simply arbitrary constants. We call them the coefficients of the polynomial.

The degree of a polynomial is the highest power of x that appears, which is n for polynomials of the above form.

Degree of a product of polynomials

AHL 2.12

The degree of a product of polynomials is the sum of the degrees:

(anxn+⋯)×(bmxm+⋯)=anbmxn+m

Specific terms & coefficients of polynomials

AHL 2.12

Each term akxk in a polynomial is associated with a specific power of x. We say that

akxk is the xk term

and

ak is the coefficient of the xk term

Specific term in a product of polynomials

AHL 2.12

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.

Matching terms in polynomial equations

AHL 2.12

If two polynomials are equal for all x∈R (not just an intersection), then all their coefficients are equal:

k=0∑nakxk≡k=0∑nbkxk⇔ak=bk for all k∈R

Polynomial division, factors and remainders

5 skills

Polynomial division: divisor, quotient and remainder.

AHL 2.12

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:

P(x)=D(x)Q(x)+R(x)🚫

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).

Polynomial division with linear divisor

AHL 2.12

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.

+00x2+11x+029x−5)−11x2−26x−135−11x2+55x−13529x−135−29x+145110

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.

Degrees of quotient and remainder

AHL 2.12

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

P(x)=Q(x)D(x)+R(x)

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 🚫.

Factor theorem

AHL 2.12

The polynomial factor theorem states that

(x−a) is a factor of P(x)⇔P(a)=0🚫

Remainder theorem

AHL 2.12

The remainder theorem states that

the emainder when P(x) is divided by (x−a) isP(a)🚫

Polynomial graphs

2 skills

Positive and negative polynomials

AHL 2.12

The coefficient a of the leading term (the one with the highest ) determines the broad shape of its graph:

Cubic

Quartic

Roots & x-intercepts

AHL 2.12

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:

Polynomial roots

3 skills

Degree n polynomial has n roots

AHL 2.12

A polynomial with degree n has n roots. These roots are not necessarily distinct, and not always real.

Sum and product of roots

AHL 2.12

For a polynomial's roots:

thesumistheproductisan−an−1an(−1)na0

Conjugate root theorem

AHL 2.12

For a polynomial with real coefficients, if a complex number z is a root, then the conjugate z⋆ must also be a root.

Skill Checklist

Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.

15 Skills Available

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

Polynomial Basics

5 skills

Polynomial definition and degree

AHL 2.12

A polynomial is a combination of positive integer powers xn:

P(x)=anxn+an−1xn−1+⋯+a1x+a0

This can also be expressed as a summation:

P(x)=k=0∑nakxk

All the afactors are simply arbitrary constants. We call them the coefficients of the polynomial.

The degree of a polynomial is the highest power of x that appears, which is n for polynomials of the above form.

Degree of a product of polynomials

AHL 2.12

The degree of a product of polynomials is the sum of the degrees:

(anxn+⋯)×(bmxm+⋯)=anbmxn+m

Specific terms & coefficients of polynomials

AHL 2.12

Each term akxk in a polynomial is associated with a specific power of x. We say that

akxk is the xk term

and

ak is the coefficient of the xk term

Specific term in a product of polynomials

AHL 2.12

Matching terms in polynomial equations

AHL 2.12

If two polynomials are equal for all x∈R (not just an intersection), then all their coefficients are equal:

k=0∑nakxk≡k=0∑nbkxk⇔ak=bk for all k∈R

Polynomial division, factors and remainders

5 skills

Polynomial division: divisor, quotient and remainder.

AHL 2.12

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:

P(x)=D(x)Q(x)+R(x)🚫

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).

Polynomial division with linear divisor

AHL 2.12

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.

+00x2+11x+029x−5)−11x2−26x−135−11x2+55x−13529x−135−29x+145110

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.

Degrees of quotient and remainder

AHL 2.12

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

P(x)=Q(x)D(x)+R(x)

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 🚫.

Factor theorem

AHL 2.12

The polynomial factor theorem states that

(x−a) is a factor of P(x)⇔P(a)=0🚫

Remainder theorem

AHL 2.12

The remainder theorem states that

the emainder when P(x) is divided by (x−a) isP(a)🚫

Polynomial graphs

2 skills

Positive and negative polynomials

AHL 2.12

The coefficient a of the leading term (the one with the highest ) determines the broad shape of its graph:

Cubic

Quartic

Roots & x-intercepts

AHL 2.12

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:

Polynomial roots

3 skills

Degree n polynomial has n roots

AHL 2.12

A polynomial with degree n has n roots. These roots are not necessarily distinct, and not always real.

Sum and product of roots

AHL 2.12

For a polynomial's roots:

thesumistheproductisan−an−1an(−1)na0

Conjugate root theorem

AHL 2.12

For a polynomial with real coefficients, if a complex number z is a root, then the conjugate z⋆ must also be a root.