A-level Mathematics C2 Adding and Factoring Polynomials

    Divide $x^3\; +\; 8x^2\; -\; 4x\; +\; 10$ by $x^2\; +\; 3x\; -1$.

    The first move is to set up the equation. Create abiding that it is in adjustment from accomplished ability of x to everyman ability of x.

    Then we bisect the first appellation of the allotment by the first appellation of the divisor.

    $frac\; =\; x$

    We abode this resultant on top.

    Then we accumulate the resultant by the divisor and decrease it from the dividend.

    $x\; left\; (\; x^2\; +\; 3x\; +1$
ight )= x^3 + 3x^2 -x

    What is larboard becomes the new allotment and we echo the action again. We abide to do this until the first appellation has a amount beneath than the first appellation of the divisor. What is larboard is the remainder.

    Synthetic analysis is one of the easiest way to bisect polynomials. It alone works if the divisor has the anatomy x - c. Agenda if the blueprint is x + c then you charge to abate c: x - (-c). In adjustment to authenticate how constructed analysis works actuality is a problem.

    Divide $2x^5\; +\; 5x^2\; -\; 10x^3\; -30x\; -\; 171$ by x-3.

    The divisor is c. So in this case it will be 3.

    Then we charge to align our divisor in adjustment from accomplished to lowest, and replacing any missing amount with with $+\; 0x^$.

    $2x^5\; +\; 0x^4\; -\; 10x^3\; +\; 5x^2\; -30x\; -\; 171$

    Then to set up the blueprint for constructed analysis we charge to abolish all the variables.

    2 + 0 - 10 + 5 - 30 - 171

    Now we set up our analysis equation.

    Next we backpack up the first appellation of the divisior.

    Then we accumulate the resultant by the divisor and add it to the next term.

    We abide to do this until we ability the end.

    Now we charge to readd the variables. If we readd the variables we go from accomplished amount -1 to everyman degree. The endure amount is the remainder.

    $2x^4\; +\; 6x^3\; +\; 8x^2\; +\; 29x\; +57$ butt 0.

    This is the acknowledgment to the problem.

    The butt theoreom states that: If you accept a polynomial f(x) disconnected by x - c, the butt is according to f(c). Agenda if the blueprint is x + c then you charge to abate c: f(-c). Actuality is an example.

    What will the butt be if you bisect $x^3\; +\; 8x^2\; -\; 4x^2\; +\; 17x\; -\; 40$ is disconnected by x - 3?

    $f(3)=\; 3^3\; +\; 8\; left\; (\; 3$
ight )^2 - 4left ( 3
ight )^2 + 17left ( 3
ight ) - 40 = 46

    The butt is 46.

    When you agency an blueprint you try to unmultiply the equation. The N-Roots Assumption states that if f(x) is a polynomial of amount greater than or according to 1, then f(x) has absolutely n roots, accouterment that a basis of multiplcity k is counted k times. The endure allotment agency that if an blueprint has 2 roots that are both 6, then we calculation 6 as 2 roots.

    The agency assumption allows us to analysis whether a amount is a factor. It states:

    A polynomial f(x) has a agency x - c if and alone if f(c) = 0. In example:

    Determine if x + 2 is a agency of $2x^2\; +\; 3x\; -2$.

    Since c is absolute instead of abrogating we charge to use this basal identity:

    $x\; +\; 2\; =\; x\; -\; left\; (\; -\; 2$
ight )

    Now we can use the agency theorem.

    $2\; left\; (-2$
ight )^2 + 3 left (-2
ight ) -2 = 8 - 6 - 2 = 0.

    Since the resultant is 0, -2 is a agency of $2x^2\; +\; 3x\; -2$.