In a platform video game, the height of a character who jumps follows a curve: it rises, reaches the top and comes back down to the ground. The moments when the character touches the ground are exactly the points where that curve crosses the horizontal axis. If the height is given by a polynomial $P(x)$, those moments are its roots: the values of $x$ for which $P(x) = 0$. This section is about finding them without having to draw the graph every time.
Justify, without performing any complex calculation or using Ruffini's rule, why $x = 0$ must necessarily be a root of the polynomial $V(x) = 5x^5 - 7x^3 + 2x^2 - 9x$.
Justify, without performing any complex calculation or using Ruffini's rule, why $x = 0$ must necessarily be a root of the polynomial $V(x) = 5x^5 - 7x^3 + 2x^2 - 9x$.
If you plot $y = P(x)$, its roots are exactly the points where the curve touches or crosses the $X$-axis. That is why they are sometimes also called the "zeros" of the polynomial.
Look at the graph: there are three points marked on the $X$-axis, and these are the three values of $x$ that make the polynomial equal to zero. Notice too that a double root (the polynomial touches the axis without crossing it, like a root that is repeated twice) looks different from a simple one: the curve "bounces" instead of passing through.
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Observe the graph of the polynomial function $y = P(x)$ and write down its real roots based on the points of intersection with the axes.
Observe the graph of the polynomial function $y = P(x)$ and write down its real roots based on the points of intersection with the axes.
Ruffini: dividing by $(x-a)$ and checking the remainder
Substituting works well to check a candidate, but when you already know (or suspect) that $a$ is a root, Ruffini's rule also gives you the quotient, that is, the polynomial that is left after "removing" that root.
If the remainder when $P(x)$ is divided by $(x-a)$ is $0$, then $a$ is a root and $(x-a)$ is a factor of $P(x)$. If the remainder is not $0$, $a$ is not a root: there is no need to try anything else with that number.
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Use Ruffini's rule to check whether $x = 3$ is a root of the polynomial $Q(x) = 2x^4 - 5x^3 - 4x^2 + 7x - 6$.
Use Ruffini's rule to check whether $x = 3$ is a root of the polynomial $Q(x) = 2x^4 - 5x^3 - 4x^2 + 7x - 6$.
Candidates for integer roots: the factors of the constant term
Trying numbers at random is inefficient. If the polynomial has integer coefficients, its integer roots (if it has any) are always among the factors of the constant term.
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Write down all the integers that are candidates for being roots of the monic polynomial $R(x) = x^4 + 3x^3 - 5x^2 + x - 12$, based on its constant term.
Write down all the integers that are candidates for being roots of the monic polynomial $R(x) = x^4 + 3x^3 - 5x^2 + x - 12$, based on its constant term.
Determine all rational roots of the polynomial $R(x) = 3x^3 + 5x^2 - 4x - 4$. To do so, first state the complete list of candidate fractions and then check which ones…
Determine all rational roots of the polynomial $R(x) = 3x^3 + 5x^2 - 4x - 4$. To do so, first state the complete list of candidate fractions and then check which ones make the polynomial zero.
If you know all the roots of a polynomial, you can write it directly as a product of factors $(x - \text{root})$, multiplied by the leading coefficient. This is the idea that links roots with the factorisation you will study in depth with common factors and special identities in other sections.
This process relies on common factors and special identities to simplify before using Ruffini's rule: both tools are covered in detail in the sections on factorising polynomials.
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Construct a cubic polynomial, expanded and in descending powers, that has $x = 1$, $x = -3$ and $x = 4$ as exact roots, and whose leading coefficient is $1$.
Construct a cubic polynomial, expanded and in descending powers, that has $x = 1$, $x = -3$ and $x = 4$ as exact roots, and whose leading coefficient is $1$.
Write a quartic polynomial, in expanded form, that has $x = 2$ as a double root, $x = -1$ as a simple root, has no constant term, and whose leading coefficient is $3$.
Write a quartic polynomial, in expanded form, that has $x = 2$ as a double root, $x = -1$ as a simple root, has no constant term, and whose leading coefficient is $3$.
Knowing that a monic cubic polynomial $P(x) = x^3 + ax^2 + bx + c$ has roots $x_1 = -1$, $x_2 = 2$ and $x_3 = 4$, determine the numerical value of its constant term $c$…
Knowing that a monic cubic polynomial $P(x) = x^3 + ax^2 + bx + c$ has roots $x_1 = -1$, $x_2 = 2$ and $x_3 = 4$, determine the numerical value of its constant term $c$ and its degree-two coefficient $a$. Justify your answer by deducing it from the factorised form without performing the full multiplication of the three binomials.