Polynomial Functions and Their Graphs Practice

Ignore term count and coefficient size; identify the greatest exponent with a nonzero coefficient.

The greatest exponent with a nonzero coefficient in \(7x^5-2x^2+9\) is \(5\), so the degree is \(5\).

Ignore term count and coefficient size; identify the greatest exponent with a nonzero coefficient.

The greatest exponent with a nonzero coefficient in \(-3x^4+x^3-8\) is \(4\), so the degree is \(4\).

Ignore term count and coefficient size; identify the greatest exponent with a nonzero coefficient.

The greatest exponent with a nonzero coefficient in \(11-6x+x^2\) is \(2\), so the degree is \(2\).

Ignore term count and coefficient size; identify the greatest exponent with a nonzero coefficient.

The greatest exponent with a nonzero coefficient in \(9x-14\) is \(1\), so the degree is \(1\).

Ignore term count and coefficient size; identify the greatest exponent with a nonzero coefficient.

The greatest exponent with a nonzero coefficient in \(-12\) is \(0\), so the degree is \(0\).

Use the degree as the maximum possible real-zero count; do not assume every zero is real.

A polynomial's number of real zeros cannot exceed its degree, so the maximum possible count is \(1\). This is an upper bound, not a guarantee.

Use the degree as the maximum possible real-zero count; do not assume every zero is real.

A polynomial's number of real zeros cannot exceed its degree, so the maximum possible count is \(2\). This is an upper bound, not a guarantee.

Use the degree as the maximum possible real-zero count; do not assume every zero is real.

A polynomial's number of real zeros cannot exceed its degree, so the maximum possible count is \(3\). This is an upper bound, not a guarantee.

Use the degree as the maximum possible real-zero count; do not assume every zero is real.

A polynomial's number of real zeros cannot exceed its degree, so the maximum possible count is \(4\). This is an upper bound, not a guarantee.

Use the degree as the maximum possible real-zero count; do not assume every zero is real.

A polynomial's number of real zeros cannot exceed its degree, so the maximum possible count is \(6\). This is an upper bound, not a guarantee.

Use parentheses around negative inputs and evaluate powers before multiplication and addition.

Substitute \(x=2\) into every power of the polynomial. Careful evaluation gives \(f(2)=6\).

Use parentheses around negative inputs and evaluate powers before multiplication and addition.

Substitute \(x=-1\) into every power of the polynomial. Careful evaluation gives \(f(-1)=-6\).

Use parentheses around negative inputs and evaluate powers before multiplication and addition.

Substitute \(x=-2\) into every power of the polynomial. Careful evaluation gives \(f(-2)=6\).

Use parentheses around negative inputs and evaluate powers before multiplication and addition.

Substitute \(x=3\) into every power of the polynomial. Careful evaluation gives \(f(3)=15\).

Use parentheses around negative inputs and evaluate powers before multiplication and addition.

Substitute \(x=4\) into every power of the polynomial. Careful evaluation gives \(f(4)=0\).

Read horizontal coordinates of axis intersections; do not report vertical coordinates.

Zeros are horizontal coordinates where the curve meets the horizontal axis. The generated curve crosses or touches at \(-3,0,2\), and each value makes its factored polynomial equal zero.

Read horizontal coordinates of axis intersections; do not report vertical coordinates.

Zeros are horizontal coordinates where the curve meets the horizontal axis. The generated curve crosses or touches at \(-2,3\), and each value makes its factored polynomial equal zero.

Read horizontal coordinates of axis intersections; do not report vertical coordinates.

Zeros are horizontal coordinates where the curve meets the horizontal axis. The generated curve crosses or touches at \(-3,-1,1,3\), and each value makes its factored polynomial equal zero.

Read horizontal coordinates of axis intersections; do not report vertical coordinates.

Zeros are horizontal coordinates where the curve meets the horizontal axis. The generated curve crosses or touches at \(-2,1,4\), and each value makes its factored polynomial equal zero.

Read horizontal coordinates of axis intersections; do not report vertical coordinates.

Zeros are horizontal coordinates where the curve meets the horizontal axis. The generated curve crosses or touches at \(-1\), and each value makes its factored polynomial equal zero.

Use the equation's greatest exponent; graph crossings alone may not reveal every degree detail.

The displayed equation has greatest power \(3\). Its 3 visible real zeros do not change that algebraic degree.

Use the equation's greatest exponent; graph crossings alone may not reveal every degree detail.

The displayed equation has greatest power \(2\). Its 2 visible real zeros do not change that algebraic degree.

Use the equation's greatest exponent; graph crossings alone may not reveal every degree detail.

The displayed equation has greatest power \(4\). Its 4 visible real zeros do not change that algebraic degree.

Use the equation's greatest exponent; graph crossings alone may not reveal every degree detail.

The displayed equation has greatest power \(3\). Its 3 visible real zeros do not change that algebraic degree.

Use the equation's greatest exponent; graph crossings alone may not reveal every degree detail.

The displayed equation has greatest power \(1\). Its 1 visible real zero do not change that algebraic degree.

Translate a zero c into the factor x-c, keeping the sign reversal visible.

The graph has an intercept at horizontal coordinate \(-3\), so \(f(-3)=0\). The corresponding factor is \(x-(-3)=x+3\).

Translate a zero c into the factor x-c, keeping the sign reversal visible.

The graph has an intercept at horizontal coordinate \(-2\), so \(f(-2)=0\). The corresponding factor is \(x-(-2)=x+2\).

Translate a zero c into the factor x-c, keeping the sign reversal visible.

The graph has an intercept at horizontal coordinate \(-3\), so \(f(-3)=0\). The corresponding factor is \(x-(-3)=x+3\).

Translate a zero c into the factor x-c, keeping the sign reversal visible.

The graph has an intercept at horizontal coordinate \(-2\), so \(f(-2)=0\). The corresponding factor is \(x-(-2)=x+2\).

Translate a zero c into the factor x-c, keeping the sign reversal visible.

The graph has an intercept at horizontal coordinate \(-1\), so \(f(-1)=0\). The corresponding factor is \(x-(-1)=x+1\).

Read the curve from left to right and distinguish a point's horizontal input from its vertical function value.

The curve changes from increasing to decreasing at the labeled point (-1,2), so the requested input is -1.

Read the curve from left to right and distinguish a point's horizontal input from its vertical function value.

The local maximum point is (-1,2); the maximum value is its vertical coordinate 2.

Read the curve from left to right and distinguish a point's horizontal input from its vertical function value.

The curve changes from decreasing to increasing at the labeled point (1,-2), so the requested input is 1.

Read the curve from left to right and distinguish a point's horizontal input from its vertical function value.

From left to right, the curve falls between its local maximum at x=-1 and local minimum at x=1.

Read the curve from left to right and distinguish a point's horizontal input from its vertical function value.

The curve rises before x=-1, falls until x=1, and rises again after x=1.

Ignore points outside the stated interval and report the vertical value when the question asks for a value.

Within the stated closed interval, compare the endpoints and the parabola's turning point. The requested minimum value is \(-4\), reached at \(x=0\).

Ignore points outside the stated interval and report the vertical value when the question asks for a value.

Within the stated closed interval, compare the endpoints and the parabola's turning point. The requested maximum value is \(5\), reached at \(x=2\).

Ignore points outside the stated interval and report the vertical value when the question asks for a value.

Within the stated closed interval, compare the endpoints and the parabola's turning point. The requested minimum value is \(-4\), reached at \(x=1\).

Ignore points outside the stated interval and report the vertical value when the question asks for a value.

Within the stated closed interval, compare the endpoints and the parabola's turning point. The requested minimum value is \(-4\), reached at \(x=-2\).

Ignore points outside the stated interval and report the vertical value when the question asks for a value.

Within the stated closed interval, compare the endpoints and the parabola's turning point. The requested minimum value is \(-4\), reached at \(x=2\).

Translate every listed zero independently into x-c and multiply the factors without changing their signs.

Each zero \(c\) contributes factor \(x-c\). Applying that rule to \(-4,2\) gives \((x+4)(x-2)\).

Translate every listed zero independently into x-c and multiply the factors without changing their signs.

Each zero \(c\) contributes factor \(x-c\). Applying that rule to \(-3,1,5\) gives \((x+3)(x-1)(x-5)\).

Translate every listed zero independently into x-c and multiply the factors without changing their signs.

Each zero \(c\) contributes factor \(x-c\). Applying that rule to \(-2,0,3\) gives \((x+2)x(x-3)\).

Translate every listed zero independently into x-c and multiply the factors without changing their signs.

Each zero \(c\) contributes factor \(x-c\). Applying that rule to \(-1,2,4,6\) gives \((x+1)(x-2)(x-4)(x-6)\).

Translate every listed zero independently into x-c and multiply the factors without changing their signs.

Each zero \(c\) contributes factor \(x-c\). Applying that rule to \(0,3\) gives \(x(x-3)\).

Turn the x-intercept into a zero function value before solving the parameter equation.

An intercept at \(x=2\) means \(p(2)=0\). The nonparameter terms evaluate to \(-2\), so \(-2+k=0\), giving \(k=2\).

Turn the x-intercept into a zero function value before solving the parameter equation.

An intercept at \(x=-1\) means \(p(-1)=0\). The nonparameter terms evaluate to \(-3\), so \(-3+k=0\), giving \(k=3\).

Turn the x-intercept into a zero function value before solving the parameter equation.

An intercept at \(x=3\) means \(p(3)=0\). The nonparameter terms evaluate to \(5\), so \(5+2k=0\), giving \(k=-2.5\).

Turn the x-intercept into a zero function value before solving the parameter equation.

An intercept at \(x=-2\) means \(p(-2)=0\). The nonparameter terms evaluate to \(-5\), so \(-5-k=0\), giving \(k=-5\).

Turn the x-intercept into a zero function value before solving the parameter equation.

An intercept at \(x=4\) means \(p(4)=0\). The nonparameter terms evaluate to \(-2\), so \(-2+k=0\), giving \(k=2\).